\newpage \section{Examples} % (fold) \label{sec:examples} \subsection{D'Alembert 1} % (fold) \label{sub:d_alembert_1} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.A = point : new (0,0) z.a = point : new (4,0) z.B = point : new (7,-1) z.b = point : new (5.5,-1) z.C = point : new (5,-4) z.c = point : new (4.25,-4) C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) C.Cc = circle : new (z.C,z.c) z.I = C.Aa : external_similitude (C.Bb) z.J = C.Aa : external_similitude (C.Cc) z.K = C.Cc : external_similitude (C.Bb) z.Ip = C.Aa : internal_similitude (C.Bb) z.Jp = C.Aa : internal_similitude (C.Cc) z.Kp = C.Cc : internal_similitude (C.Bb) \end{tkzelements} \begin{tikzpicture}[rotate=-60] \tkzGetNodes \tkzDrawCircles(A,a B,b C,c) \tkzDrawPoints(A,B,C,I,J,K,I',J',K') \tkzDrawSegments[new](I,K A,I A,J B,I B,K C,J C,K) \tkzDrawSegments[purple](I,J' I',J I',K) \tkzLabelPoints(I,J,K,I',J',K') \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .5 z.A = point : new (0,0) z.a = point : new (4,0) z.B = point : new (7,-1) z.b = point : new (5.5,-1) z.C = point : new (5,-4) z.c = point : new (4.25,-4) C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) C.Cc = circle : new (z.C,z.c) z.I = C.Aa : external_similitude (C.Bb) z.J = C.Aa : external_similitude (C.Cc) z.K = C.Cc : external_similitude (C.Bb) z.Ip = C.Aa : internal_similitude (C.Bb) z.Jp = C.Aa : internal_similitude (C.Cc) z.Kp = C.Cc : internal_similitude (C.Bb) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture}[rotate=30] \tkzGetNodes \tkzDrawCircles(A,a B,b C,c) \tkzDrawPoints(A,B,C,I,J,K,I',J',K') \tkzDrawSegments[new](I,K A,I A,J B,I B,K C,J C,K) \tkzDrawSegments[purple](I,J' I',J I',K) \tkzLabelPoints(I,J,K,I',J',K') \end{tikzpicture} \end{minipage} % subsection d_alembert_1 (end) \subsection{D'Alembert 2} % (fold) \label{sub:d_alembert_2} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = .75 z.A = point : new (0,0) z.a = point : new (5,0) z.B = point : new (7,-1) z.b = point : new (3,-1) z.C = point : new (5,-4) z.c = point : new (2,-4) C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) C.Cc = circle : new (z.C,z.c) z.i,z.j = get_points (C.Aa : radical_axis (C.Bb)) z.k,z.l = get_points (C.Aa : radical_axis (C.Cc)) z.m,z.n = get_points (C.Bb : radical_axis (C.Cc)) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,a B,b C,c) \tkzDrawLines[new](i,j k,l m,n) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .5 z.A = point : new (0,0) z.a = point : new (5,0) z.B = point : new (7,-1) z.b = point : new (3,-1) z.C = point : new (5,-4) z.c = point : new (2,-4) C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) C.Cc = circle : new (z.C,z.c) z.i,z.j = get_points (C.Aa : radical_axis (C.Bb)) z.k,z.l = get_points (C.Aa : radical_axis (C.Cc)) z.m,z.n = get_points (C.Bb : radical_axis (C.Cc)) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,a B,b C,c) \tkzDrawLines[new](i,j k,l m,n) \end{tikzpicture} \end{minipage} % subsection d_alembert_2 (end) \subsection{Alternate} % (fold) \label{sub:alternate} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (1 , 5) T = triangle: new (z.A,z.B,z.C) z.I = T.incenter L.AI = line: new (z.A,z.I) z.D = intersection (L.AI,T.bc) L.LLC = T.ab: ll_from (z.C) z.E = intersection (L.AI,L.LLC) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[purple](C,E) \tkzDrawSegment[purple](A,E) \tkzFillAngles[purple!30,opacity=.4](B,A,C C,E,D) \tkzMarkAngles[mark=|](B,A,D D,A,C C,E,D) \tkzDrawPoints(A,...,E) \tkzLabelPoints(A,B) \tkzLabelPoints[above](C,D,E) \tkzMarkSegments(A,C C,E) \end{tikzpicture} \end{Verbatim} \end{minipage} \hspace*{\fill} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (1 , 5) T = triangle: new (z.A,z.B,z.C) z.I = T.incenter L.AI = line: new (z.A,z.I) z.D = intersection (L.AI,T.bc) L.LLC = T.ab: ll_from (z.C) z.E = intersection (L.AI,L.LLC) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[purple](C,E) \tkzDrawSegment[purple](A,E) \tkzFillAngles[purple!30,opacity=.4](B,A,C C,E,D) \tkzMarkAngles[mark=|](B,A,D D,A,C C,E,D) \tkzDrawPoints(A,...,E) \tkzLabelPoints(A,B) \tkzLabelPoints[above](C,D,E) \tkzMarkSegments(A,C C,E) \end{tikzpicture} \end{minipage} % subsection alternate (end) \subsection{Apollonius circle} % (fold) \label{sub:apollonius_circle} \begin{tkzelements} scale=.75 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.M = point: new (5 , 3) T.MAB = triangle : new (z.M,z.A,z.B) L.bis = T.MAB : bisector () z.C = L.bis.pb L.bisext = T.MAB : bisector_ext () z.D = intersection (T.MAB.bc, L.bisext) L.CD = line: new (z.C,z.D) z.O = L.CD.mid L.AM = T.MAB.ab z.E = z.M : symmetry (z.A) \end{tkzelements} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} scale=.75 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.M = point: new (5 , 3) T.MAB = triangle : new (z.M,z.A,z.B) L.bis = T.MAB : bisector () z.C = L.bis.pb L.bisext = T.MAB : bisector_ext () z.D = intersection (T.MAB.bc, L.bisext) L.CD = line: new (z.C,z.D) z.O = L.CD.mid L.AM = T.MAB.ab z.E = z.M : symmetry (z.A) \end{tkzelements} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \hspace*{\fill} \begin{tikzpicture}[scale=.8] \tkzGetNodes \tkzDrawSegment[add=0 and 1](A,M) \tkzDrawSegments[purple](M,C M,D) \tkzDrawCircle[purple](O,C) \tkzDrawSegments(A,B B,M D,B) \tkzDrawPoints(A,B,M,C,D) \tkzLabelPoints[below right](A,B,C,D) \tkzLabelPoints[above](M) \tkzFillAngles[opacity=.4,cyan!20](A,M,B) \tkzFillAngles[opacity=.4,purple!20](B,M,E) \tkzMarkRightAngle[opacity=.4,fill=gray!20](C,M,D) \tkzMarkAngles[mark=|](A,M,C C,M,B) \tkzMarkAngles[mark=||](B,M,D D,M,E) \end{tikzpicture} \hspace*{\fill} \end{minipage} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawSegment[add=0 and 1](A,M) \tkzDrawSegments[purple](M,C M,D) \tkzDrawCircle[purple](O,C) \tkzDrawSegments(A,B B,M D,B) \tkzDrawPoints(A,B,M,C,D) \tkzLabelPoints[below right](A,B,C,D) \tkzLabelPoints[above](M) \tkzFillAngles[opacity=.4,cyan!20](A,M,B) \tkzFillAngles[opacity=.4,purple!20](B,M,E) \tkzMarkRightAngle[opacity=.4,fill=gray!20](C,M,D) \tkzMarkAngles[mark=|](A,M,C C,M,B) \tkzMarkAngles[mark=||](B,M,D D,M,E) \end{tikzpicture} \end{Verbatim} Remark : The circle can be obtained with: |C.AB = T.MAB.bc : apollonius (length(z.M,z.A)/length(z.M,z.B))| %subsection apollonius_circle (end) \subsection{Apollonius and circle circumscribed } % (fold) \label{sub:apollonius_and_circle_circumscribed} \begin{Verbatim} \begin{tkzelements} scale =.75 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.M = point: new (5 , 4) T.AMB = triangle: new (z.A,z.M,z.B) L.AB = T.AMB.ca z.I = T.AMB.incenter L.MI = line: new (z.M,z.I) z.C = intersection (L.AB , L.MI) L.MJ = L.MI: ortho_from (z.M) z.D = intersection (L.AB , L.MJ) L.CD = line: new (z.C,z.D) z.O = L.CD.mid z.G = T.AMB.circumcenter C.GA = circle: new (z.G,z.A) C.OC = circle: new (z.O,z.C) _,z.N = intersection (C.GA , C.OC) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,M) \tkzDrawCircles[purple](O,C G,A) \tkzDrawSegments[purple](M,D) \tkzDrawSegments(D,B O,G M,C) \tkzDrawSegments[red,dashed](M,N M,O M,G) \tkzDrawPoints(A,B,M,C,D,N,O,G) \tkzLabelPoints[below right](A,B,C,D,N,O,G) \tkzLabelPoints[above](M) \tkzMarkRightAngle[opacity=.4,fill=gray!20](C,M,D) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale =.75 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.M = point: new (5 , 4) T.AMB = triangle: new (z.A,z.M,z.B) L.AB = T.AMB.ca z.I = T.AMB.incenter L.MI = line: new (z.M,z.I) z.C = intersection (L.AB , L.MI) L.MJ = L.MI: ortho_from (z.M) z.D = intersection (L.AB , L.MJ) L.CD = line: new (z.C,z.D) z.O = L.CD.mid z.G = T.AMB.circumcenter C.GA = circle: new (z.G,z.A) C.OC = circle: new (z.O,z.C) _,z.N = intersection (C.GA , C.OC) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,M) \tkzDrawCircles[purple](O,C G,A) \tkzDrawSegments[purple](M,D) \tkzDrawSegments(D,B O,G M,C) \tkzDrawSegments[red,dashed](M,N M,O M,G) \tkzDrawPoints(A,B,M,C,D,N,O,G) \tkzLabelPoints[below right](A,B,C,D,N,O,G) \tkzLabelPoints[above](M) \tkzMarkRightAngle[opacity=.4,fill=gray!20](C,M,D) \end{tikzpicture} \hspace*{\fill} % subsection apollonius_and_circle_circumscribed (end) \subsection{Apollonius circles in a triangle} % (fold) \label{sub:apollonius_circles} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (4.5 , 1) T.ABC = triangle: new (z.A,z.B,z.C) z.I = T.ABC.incenter z.O = T.ABC.circumcenter L.CI = line: new (z.C,z.I) z.Cp = intersection (T.ABC.ab , L.CI) z.x = L.CI.north_pa L.Cx = line: new (z.C,z.x) z.R = intersection (L.Cx,T.ABC.ab) L.CpR = line: new (z.Cp,z.R) z.O1 = L.CpR.mid L.AI = line: new (z.A,z.I) z.Ap = intersection (T.ABC.bc , L.AI) z.y = L.AI.north_pa L.Ay = line: new (z.A,z.y) z.S = intersection (L.Ay,T.ABC.bc) L.ApS = line: new (z.Ap,z.S) z.O2 = L.ApS.mid L.BI = line: new (z.B,z.I) z.Bp = intersection (T.ABC.ca , L.BI) z.z = L.BI.north_pa L.Bz = line: new (z.B,z.z) z.T = intersection (L.Bz,T.ABC.ca) L.Bpt = line: new (z.Bp,z.T) z.O3 = L.Bpt.mid \end{tkzelements} \end{Verbatim} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[blue!50!black](O1,C' O2,A' O3,B') \tkzDrawSegments[new](B,S C,T A,R) \tkzDrawPolygon(A,B,C) \tkzDrawPoints(A,B,C,A',B',C',O,I,R,S,T,O1,O2,O3) \tkzLabelPoints(A,B,C,A',B',C',O,I) \tkzLabelPoints(O1,O2,O3) \tkzDrawCircle[purple](O,A) \tkzDrawLine(O1,O2) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (4.5 , 1) T.ABC = triangle: new (z.A,z.B,z.C) z.I = T.ABC.incenter z.O = T.ABC.circumcenter L.CI = line: new (z.C,z.I) z.Cp = intersection (T.ABC.ab , L.CI) z.x = L.CI.north_pa L.Cx = line: new (z.C,z.x) z.R = intersection (L.Cx,T.ABC.ab) L.CpR = line: new (z.Cp,z.R) z.O1 = L.CpR.mid L.AI = line: new (z.A,z.I) z.Ap = intersection (T.ABC.bc , L.AI) z.y = L.AI.north_pa L.Ay = line: new (z.A,z.y) z.S = intersection (L.Ay,T.ABC.bc) L.ApS = line: new (z.Ap,z.S) z.O2 = L.ApS.mid L.BI = line: new (z.B,z.I) z.Bp = intersection (T.ABC.ca , L.BI) z.z = L.BI.north_pa L.Bz = line: new (z.B,z.z) z.T = intersection (L.Bz,T.ABC.ca) L.Bpt = line: new (z.Bp,z.T) z.O3 = L.Bpt.mid \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[blue!50!black](O1,C' O2,A' O3,B') \tkzDrawSegments[new](B,S C,T A,R) \tkzDrawPolygon(A,B,C) \tkzDrawPoints(A,B,C,A',B',C',O,I,R,S,T,O1,O2,O3) \tkzLabelPoints(A,B,C,A',B',C',O,I) \tkzLabelPoints(O1,O2,O3) \tkzDrawCircle[purple](O,A) \tkzDrawLine(O1,O2) \end{tikzpicture} \hspace*{\fill} Same result using the function |T.ABC.ab : apollonius (k) | \begin{Verbatim} \begin{tkzelements} scale = .75 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (4.5 , 1) T.ABC = triangle : new (z.A,z.B,z.C) z.O = T.ABC.circumcenter C.AB = T.ABC.ab : apollonius (length(z.C,z.A)/length(z.C,z.B)) z.w1,z.t1 = get_points ( C.AB ) C.AC = T.ABC.ca : apollonius (length(z.B,z.C)/length(z.B,z.A)) z.w2,z.t2 = get_points ( C.AC ) C.BC = T.ABC.bc : apollonius (length(z.A,z.B)/length(z.A,z.C)) z.w3,z.t3 = get_points ( C.BC ) \end{tkzelements} \end{Verbatim} % subsection apollonius_circles (end) \subsection{Archimedes} % (fold) \label{sub:archimedes} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.O_1 = point: new (0, 0) z.O_2 = point: new (0, 1) z.A = point: new (0, 3) z.F = point: polar (3, math.pi/6) L = line: new (z.F,z.O_1) C = circle: new (z.O_1,z.A) z.E = intersection (L,C) T = triangle: new (z.F,z.E,z.O_2) z.x = T: parallelogram () L = line: new (z.x,z.O_2) C = circle: new (z.O_2,z.A) z.C,z.D = intersection (L ,C) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O_1,A O_2,A) \tkzDrawSegments[new](O_1,A E,F C,D) \tkzDrawSegments[purple](A,E A,F) \tkzDrawPoints(A,O_1,O_2,E,F,C,D) \tkzLabelPoints(A,O_1,O_2,E,F,C,D) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} z.O_1 = point: new (0, 0) z.O_2 = point: new (0, 1) z.A = point: new (0, 3) z.F = point: polar (3, math.pi/6) L = line: new (z.F,z.O_1) C = circle: new (z.O_1,z.A) z.E = intersection (L,C) T = triangle: new (z.F,z.E,z.O_2) z.x = T: parallelogram () L = line: new (z.x,z.O_2) C = circle: new (z.O_2,z.A) z.C,z.D = intersection (L ,C) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O_1,A O_2,A) \tkzDrawSegments[new](O_1,A E,F C,D) \tkzDrawSegments[purple](A,E A,F) \tkzDrawPoints(A,O_1,O_2,E,F,C,D) \tkzLabelPoints(A,O_1,O_2,E,F,C,D) \end{tikzpicture} \end{minipage} % subsection archimedes (end) \subsection{Bankoff circle} % (fold) \label{sub:bankoff_circle} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (10 , 0) L.AB = line : new (z.A,z.B) z.C = L.AB: gold_ratio () L.AC = line : new (z.A,z.C) L.CB = line : new (z.C,z.B) z.O_0 = L.AB.mid z.O_1 = L.AC.mid z.O_2 = L.CB.mid C.O0B = circle : new (z.O_0,z.B) C.O1C = circle : new (z.O_1,z.C) C.O2C = circle : new (z.O_2,z.B) z.Pp = C.O0B : midarc (z.B,z.A) z.P = C.O1C : midarc (z.C,z.A) z.Q = C.O2C : midarc (z.B,z.C) L.O1O2 = line : new (z.O_1,z.O_2) L.O0O1 = line : new (z.O_0,z.O_1) L.O0O2 = line : new (z.O_0,z.O_2) z.M_0 = L.O1O2 : harmonic_ext (z.C) z.M_1 = L.O0O1 : harmonic_int (z.A) z.M_2 = L.O0O2 : harmonic_int (z.B) L.BP = line : new (z.B,z.P) L.AQ = line : new (z.A,z.Q) z.S = intersection (L.BP,L.AQ) L.PpO0 = line : new (z.Pp,z.O_0) L.PC = line : new (z.P,z.C) z.Ap = intersection (L.PpO0,L.PC) L.CS = line : new (z.C,z.S) C.M1A = circle : new (z.M_1,z.A) C.M2B = circle : new (z.M_2,z.B) z.P_0 = intersection (L.CS,C.O0B) z.P_1 = intersection (C.M2B,C.O1C) z.P_2 = intersection (C.M1A,C.O2C) T.P0P1P2 = triangle : new (z.P_0,z.P_1,z.P_2) z.O_4 = T.P0P1P2.circumcenter T.CP1P2 = triangle : new (z.C,z.P_1,z.P_2) z.O_5 = T.CP1P2.circumcenter \end{tkzelements} \end{Verbatim} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawSemiCircles[teal](O_0,B) \tkzDrawSemiCircles[teal,fill=teal!20,opacity=.5](O_1,C O_2,B) \tkzDrawCircle[fill=green!10](O_4,P_0) \tkzDrawCircle[purple,fill=purple!10,opacity=.5](O_5,C) \tkzDrawSegments(A,B O_0,P' B,P A,Q) \tkzDrawSegments(P,B Q,O_2 P,O_1) \tkzDrawSegments[purple](O_5,P_2 O_5,P_1 O_5,C) \tkzDrawPoints(A,B,C,P_0,P_2,P_1,O_0,O_1,O_2,O_4,O_5,Q,P,P',S) \tkzLabelPoints[below](A,B,C,O_0,O_1,O_2,P') \tkzLabelPoints[above](Q,P) \tkzLabelPoints[above right](P_0,P_2,P_1,O_5,O_4,S) \begin{scope}[font=\scriptsize] \tkzLabelCircle[above](O_1,C)(120){$(\beta)$} \tkzLabelCircle[above](O_2,B)(70){$(\gamma)$} \tkzLabelCircle[above](O_0,B)(110){$(\alpha)$} \tkzLabelCircle[left](O_4,P_2)(60){$(\delta)$} \tkzLabelCircle[left](O_5,C)(140){$(\epsilon)$} \end{scope} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = 1.25 z.A = point: new (0 , 0) z.B = point: new (10 , 0) L.AB = line : new (z.A,z.B) z.C = L.AB: gold_ratio () L.AC = line : new (z.A,z.C) L.CB = line : new (z.C,z.B) z.O_0 = L.AB.mid z.O_1 = L.AC.mid z.O_2 = L.CB.mid C.O0B = circle : new (z.O_0,z.B) C.O1C = circle : new (z.O_1,z.C) C.O2C = circle : new (z.O_2,z.B) z.Pp = C.O0B : midarc (z.B,z.A) z.P = C.O1C : midarc (z.C,z.A) z.Q = C.O2C : midarc (z.B,z.C) L.O1O2 = line : new (z.O_1,z.O_2) L.O0O1 = line : new (z.O_0,z.O_1) L.O0O2 = line : new (z.O_0,z.O_2) z.M_0 = L.O1O2 : harmonic_ext (z.C) z.M_1 = L.O0O1 : harmonic_int (z.A) z.M_2 = L.O0O2 : harmonic_int (z.B) L.BP = line : new (z.B,z.P) L.AQ = line : new (z.A,z.Q) z.S = intersection (L.BP,L.AQ) L.PpO0 = line : new (z.Pp,z.O_0) L.PC = line : new (z.P,z.C) z.Ap = intersection (L.PpO0,L.PC) L.CS = line : new (z.C,z.S) C.M1A = circle : new (z.M_1,z.A) C.M2B = circle : new (z.M_2,z.B) z.P_0 = intersection (L.CS,C.O0B) z.P_1 = intersection (C.M2B,C.O1C) z.P_2 = intersection (C.M1A,C.O2C) T.P0P1P2 = triangle : new (z.P_0,z.P_1,z.P_2) z.O_4 = T.P0P1P2.circumcenter T.CP1P2 = triangle : new (z.C,z.P_1,z.P_2) z.O_5 = T.CP1P2.circumcenter \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawSemiCircles[teal](O_0,B) \tkzDrawSemiCircles[teal,fill=teal!20,opacity=.5](O_1,C O_2,B) \tkzDrawCircle[fill=green!10](O_4,P_0) \tkzDrawCircle[purple,fill=purple!10,opacity=.5](O_5,C) \tkzDrawSegments(A,B O_0,P' B,P A,Q) \tkzDrawSegments(P,B Q,O_2 P,O_1) \tkzDrawSegments[purple](O_5,P_2 O_5,P_1 O_5,C) \tkzDrawPoints(A,B,C,P_0,P_2,P_1,O_0,O_1,O_2,O_4,O_5,Q,P,P',S) \tkzLabelPoints[below](A,B,C,O_0,O_1,O_2,P') \tkzLabelPoints[above](Q,P) \tkzLabelPoints[above right](P_0,P_2,P_1,O_5,O_4,S) \begin{scope}[font=\scriptsize] \tkzLabelCircle[above](O_1,C)(120){$(\beta)$} \tkzLabelCircle[above](O_2,B)(70){$(\gamma)$} \tkzLabelCircle[above](O_0,B)(110){$(\alpha)$} \tkzLabelCircle[left](O_4,P_2)(60){$(\delta)$} \tkzLabelCircle[left](O_5,C)(140){$(\epsilon)$} \end{scope} \end{tikzpicture} \hspace*{\fill} % subsection bankoff_circle (end) \subsection{Excircles} % (fold) \label{sub:excircles} \begin{Verbatim} \begin{tkzelements} scale = 0.7 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (.8,4) T = triangle: new ( z.A, z.B, z.C) z.K = T.centroid z.J_a,z.J_b,z.J_c = get_points (T: excentral()) z.T_a,z.T_b,z.T_c = get_points (T: extouch()) la = line: new ( z.A, z.T_a) lb = line: new ( z.B, z.T_b) z.G = intersection (la,lb) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints[new](J_a,J_b,J_c) \tkzClipBB \tkzDrawCircles[gray](J_a,T_a J_b,T_b J_c,T_c) \tkzDrawLines[add=1 and 1](A,B B,C C,A) \tkzDrawSegments[new](A,T_a B,T_b C,T_c) \tkzDrawSegments[new](J_a,T_a J_b,T_b J_c,T_c) \tkzDrawPolygon(A,B,C) \tkzDrawPolygon[new](T_a,T_b,T_c) \tkzDrawPoints(A,B,C,K) \tkzDrawPoints[new](T_a,T_b,T_c) \tkzLabelPoints[below left](A) \tkzLabelPoints[below](B) \tkzLabelPoints[above](C) \tkzLabelPoints[new,below left](T_b) \tkzLabelPoints[new,below right](T_c) \tkzLabelPoints[new,right=6pt](T_a) \tkzMarkRightAngles[fill=gray!15](J_a,T_a,B J_b,T_b,C J_c,T_c,A) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale=0.7 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (.8,4) T = triangle: new ( z.A, z.B, z.C) z.K = T.centroid z.J_a,z.J_b,z.J_c = get_points (T: excentral()) z.T_a,z.T_b,z.T_c = get_points (T: extouch()) la = line: new ( z.A, z.T_a) lb = line: new ( z.B, z.T_b) z.G = intersection (la,lb) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints[new](J_a,J_b,J_c) \tkzClipBB \tkzDrawCircles[gray](J_a,T_a J_b,T_b J_c,T_c) \tkzDrawLines[add=1 and 1](A,B B,C C,A) \tkzDrawSegments[new](A,T_a B,T_b C,T_c) \tkzDrawSegments[new](J_a,T_a J_b,T_b J_c,T_c) \tkzDrawPolygon(A,B,C) \tkzDrawPolygon[new](T_a,T_b,T_c) \tkzDrawPoints(A,B,C,K) \tkzDrawPoints[new](T_a,T_b,T_c) \tkzLabelPoints[below left](A) \tkzLabelPoints[below](B) \tkzLabelPoints[above](C) \tkzLabelPoints[new,below left](T_b) \tkzLabelPoints[new,below right](T_c) \tkzLabelPoints[new,right=6pt](T_a) \tkzMarkRightAngles[fill=gray!15](J_a,T_a,B J_b,T_b,C J_c,T_c,A) \end{tikzpicture} \hspace*{\fill} % subsection excircles (end) \subsection{Orthogonal circle through} % (fold) \label{sub:orthogonal_circle_through} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.O = point: new (0,1) z.A = point: new (1,0) z.z1 = point: new (-1.5,-1.5) z.z2 = point: new (2.5,-1.25) C.OA = circle: new (z.O,z.A) C = C.OA: orthogonal_through (z.z1,z.z2) z.c = C.center \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzDrawCircle[new](c,z1) \tkzDrawPoints[new](O,A,z1,z2,c) \tkzLabelPoints[right](O,A,z1,z2,c) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth}\begin{tkzelements} z.O = point: new (0,1) z.A = point: new (1,0) z.z1 = point: new (-1.5,-1.5) z.z2 = point: new (2.5,-1.25) C.OA = circle: new (z.O,z.A) C = C.OA: orthogonal_through (z.z1,z.z2) z.c = C.center \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzDrawCircle[orange](c,z1) \tkzDrawPoints[orange](O,A,z1,z2,c) \tkzLabelPoints[right](O,A,z1,z2,c) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection orthogonal_circle_through (end) \subsection{Divine ratio} % (fold) \label{sub:divine_ratio} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (8 , 0) L.AB = line: new (z.A,z.B) z.C = L.AB: gold_ratio () L.AC = line: new (z.A,z.C) z.O_1 = L.AC.mid _,_,z.G,z.H = get_points(L.AB: square ()) _,_,z.E,z.F = get_points(L.AC: square ()) L.CB = line: new (z.C,z.B) z.O_2 = L.CB.mid z.O_0 = L.AB.mid L.BE = line: new (z.B,z.E) L.GH = line: new (z.G,z.H) z.K = intersection (L.BE,L.GH) C0 = circle: new (z.O_0,z.B) z.R,_ = intersection (L.BE,C0) C2 = circle: new (z.O_2,z.B) z.S,_ = intersection (L.BE,C2) L.AR = line: new (z.A,z.R) C1 = circle: new (z.O_1,z.C) _,z.T = intersection (L.AR,C1) L.BG = line: new (z.B,z.G) z.L = intersection (L.AR,L.BG) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(A,C,E,F A,B,G,H) \tkzDrawCircles(O_1,C O_2,B O_0,B) \tkzDrawSegments(H,C B,K A,L) \tkzDrawPoints(A,B,C,K,E,F,G,H,O_0,O_1,O_2,R,S,T,L) \tkzLabelPoints(A,B,C,K,E,F,G,H,O_0,O_1,O_2,R,S,T,L) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (8 , 0) L.AB = line: new (z.A,z.B) z.C = L.AB: gold_ratio () L.AC = line: new (z.A,z.C) z.O_1 = L.AC.mid _,_,z.G,z.H = get_points(L.AB: square ()) _,_,z.E,z.F = get_points(L.AC: square ()) L.CB = line: new (z.C,z.B) z.O_2 = L.CB.mid z.O_0 = L.AB.mid L.BE = line: new (z.B,z.E) L.GH = line: new (z.G,z.H) z.K = intersection (L.BE,L.GH) C0 = circle: new (z.O_0,z.B) z.R,_ = intersection (L.BE,C0) C2 = circle: new (z.O_2,z.B) z.S,_ = intersection (L.BE,C2) L.AR = line: new (z.A,z.R) C1 = circle: new (z.O_1,z.C) _,z.T = intersection (L.AR,C1) L.BG = line: new (z.B,z.G) z.L = intersection (L.AR,L.BG) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(A,C,E,F A,B,G,H) \tkzDrawCircles(O_1,C O_2,B O_0,B) \tkzDrawSegments(H,C B,K A,L) \tkzDrawPoints(A,B,C,K,E,F,G,H,O_0,O_1,O_2,R,S,T,L) \tkzLabelPoints(A,B,C,K,E,F,G,H,O_0,O_1,O_2,R,S,T,L) \end{tikzpicture} \hspace*{\fill} % subsection divine_ratio (end) \subsection{Director circle} % (fold) \label{sub:director_circle} % modif C: point (0.25) instead of 2 \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = .5 z.O = point: new (0 , 0) z.F1 = point: new (4 , 0) z.F2 = point: new (-4 , 0) z.H = point: new (4*math.sqrt(2) , 0) E = ellipse: foci (z.F2,z.F1,z.H) a,b = E.Rx, E.Ry z.A = E.covertex T = triangle: new (z.H,z.O,z.A) z.P = T: parallelogram () C = circle: new (z.O,z.P) z.L = C: point (0.25) L.J,L.K = E: tangent_from (z.L) z.J = L.J.pb z.K = L.K.pb \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(F1,F2,O) \tkzDrawCircles[teal](O,P) \tkzDrawPolygon(H,O,A,P) \tkzDrawEllipse[red](O,\tkzUseLua{a},\tkzUseLua{b},0) \tkzDrawSegments[orange](O,P O,L L,J L,K) \tkzDrawPoints(F1,F2,O,H,A,P,L,J,K) \tkzLabelPoints(F1,F2,O,H,A,P,L,J,K) \tkzLabelPoints[above](L) \tkzMarkRightAngles(A,P,H J,L,K) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .5 z.O = point: new (0 , 0) z.F1 = point: new (4 , 0) z.F2 = point: new (-4 , 0) z.H = point: new (4*math.sqrt(2) , 0) E = ellipse: foci (z.F2,z.F1,z.H) a,b = E.Rx, E.Ry z.A = E.covertex T = triangle: new (z.H,z.O,z.A) z.P = T: parallelogram () C = circle: new (z.O,z.P) z.L = C: point (0.25) L.J,L.K = E: tangent_from (z.L) z.J = L.J.pb z.K = L.K.pb \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(F1,F2,O) \tkzDrawCircles[teal](O,P) \tkzDrawPolygon(H,O,A,P) \tkzDrawEllipse[red](O,\tkzUseLua{a},\tkzUseLua{b},0) \tkzDrawSegments[orange](O,P O,L L,J L,K) \tkzDrawPoints(F1,F2,O,H,A,P,L,J,K) \tkzLabelPoints(F1,F2,O,H,A,P,L,J,K) \tkzLabelPoints[above](L) \tkzMarkRightAngles(A,P,H J,L,K) \end{tikzpicture} \end{minipage} % subsection director_circle (end) \subsection{Gold division} % (fold) \label{sub:gold_division} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0,0) z.B = point: new (2.5,0) L.AB = line: new (z.A,z.B) C.AB = circle: new (z.A,z.B) C.BA = circle: new (z.B,z.A) z.J = L.AB: midpoint () L.JB = line:new (z.J,z.B) z.F,z.E = intersection (C.AB , C.BA) z.I,_ = intersection (L.AB , C.BA) z.K = L.JB : midpoint () L.mediator = L.JB: mediator () z.G = intersection (L.mediator,C.BA) L.EG = line:new (z.E,z.G) z.C = intersection (L.EG,L.AB) z.O = C.AB: antipode (z.B) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawArc[delta=5](O,B)(G) \tkzDrawCircles(A,B B,A) \tkzDrawSegments(A,E B,E O,I) \tkzDrawSegments[purple](J,E A,G G,I K,G E,G) \tkzMarkSegments[mark=s||](A,E B,E O,A) \tkzDrawPoints(A,B,C,E,I,J,G,O,K) \tkzLabelPoints(A,B,C,E,I,J,G,O,K) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.A = point: new (0,0) z.B = point: new (2.5,0) L.AB = line: new (z.A,z.B) C.AB = circle: new (z.A,z.B) C.BA = circle: new (z.B,z.A) z.J = L.AB: midpoint () L.JB = line:new (z.J,z.B) z.F,z.E = intersection (C.AB , C.BA) z.I,_ = intersection (L.AB , C.BA) z.K = L.JB : midpoint () L.mediator = L.JB: mediator () z.G = intersection (L.mediator,C.BA) L.EG = line:new (z.E,z.G) z.C = intersection (L.EG,L.AB) z.O = C.AB: antipode (z.B) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawArc[delta=5](O,B)(G) \tkzDrawCircles(A,B B,A) \tkzDrawSegments(A,E B,E O,I) \tkzDrawSegments[purple](J,E A,G G,I K,G E,G) \tkzMarkSegments[mark=s||](A,E B,E O,A) \tkzDrawPoints(A,B,C,E,I,J,G,O,K) \tkzLabelPoints(A,B,C,E,I,J,G,O,K) \end{tikzpicture} \hspace*{\fill} % subsection gold_division (end) \subsection{Ellipse} % (fold) \label{sub:ellipse} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.C = point: new (3 , 2) z.A = point: new (5 , 1) L.CA = line : new (z.C,z.A) z.b = L.CA.north_pa L = line : new (z.C,z.b) z.B = L : point (0.5) E = ellipse: new (z.C,z.A,z.B) a = E.Rx b = E.Ry slope = math.deg(E.slope) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal](C,A) \tkzDrawEllipse[red](C,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{slope}) \tkzDrawPoints(C,A,B,b) \tkzLabelPoints(C,A,B) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} z.C = point: new (3 , 2) z.A = point: new (5 , 1) L.CA = line : new (z.C,z.A) z.b = L.CA.north_pa L = line : new (z.C,z.b) z.B = L : point (0.5) E = ellipse: new (z.C,z.A,z.B) a = E.Rx b = E.Ry slope = math.deg(E.slope) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal](C,A) \tkzDrawEllipse[red](C,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{slope}) \tkzDrawPoints(C,A,B,b) \tkzLabelPoints(C,A,B) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection ellipse (end) \subsection{Ellipse with radii} % (fold) \label{sub:ellipse_with_radii} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} scale=.5 z.C = point: new (0 , 4) b = value(math.sqrt(8)) a = value(math.sqrt(32)) ang = math.deg(math.pi/4) E = ellipse: radii (z.C,a,b,math.pi/4) z.V = E : point (0) z.CoV = E : point (math.pi/2) \end{tkzelements} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawEllipse[blue](C,\tkzUseLua{a}, \tkzUseLua{b},\tkzUseLua{ang}) \tkzDrawPoints(C,V,CoV) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} scale=.5 z.C = point: new (0 , 4) b = value(math.sqrt(8)) a = value(math.sqrt(32)) ang = math.deg(math.pi/4) E = ellipse: radii (z.C,a,b,math.pi/4) z.V = E : point (0) z.CoV = E : point (math.pi/2) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawEllipse[blue](C,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{ang}) \tkzDrawPoints(C,V,CoV) \end{tikzpicture} \end{minipage} % subsection ellipse_with_radii (end) \subsection{Ellipse\_with\_foci} % (fold) \label{sub:ellipse_with_foci} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} local e e = .8 z.A = point: new (2 , 3) z.B = point: new (5 , 4) z.K = point: new (6, 7) L.AB = line: new (z.A,z.B) z.C = L.AB.mid c = point.abs(z.B-z.C) a = c/e b = math.sqrt (a^2-c^2) z.V = z.C + a*(z.B-z.C)/point.abs(z.B-z.C) E = ellipse: foci (z.A,z.B,z.V) z.cV = E.covertex ang = math.deg(E.slope) L.ta,L.tb = E: tangent_from (z.K) z.F = L.ta.pb z.G = L.tb.pb \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C,K,F,G,V,cV) \tkzLabelPoints(A,B,C,K,F,G,V,cV) \tkzDrawEllipse[teal](C,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{ang}) \tkzDrawLines(K,F K,G) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} local e e = .8 z.A = point: new (2 , 3) z.B = point: new (5 , 4) z.K = point: new (6, 7) L.AB = line: new (z.A,z.B) z.C = L.AB.mid c = point.abs(z.B-z.C) a = c/e b = math.sqrt (a^2-c^2) z.V = z.C + a*(z.B-z.C)/point.abs(z.B-z.C) E = ellipse: foci (z.A,z.B,z.V) z.cV = E.covertex ang = math.deg(E.slope) L.ta,L.tb = E: tangent_from (z.K) z.F = L.ta.pb z.G = L.tb.pb \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C,K,F,G,V,cV) \tkzLabelPoints(A,B,C,K,F,G,V,cV) \tkzDrawEllipse[teal](C,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{ang}) \tkzDrawLines(K,F K,G) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection ellipse_with_foci (end) \subsection{Euler relation} % (fold) \label{sub:euler_relation} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} scale = .75 z.A = point: new (0 , 0) z.B = point: new (5 , 0) z.C = point: new (-.4 , 4) T.ABC = triangle: new (z.A,z.B,z.C) z.J,z.K = get_points(T.ABC: ex_circle (2)) z.X,z.Y,z.K= T.ABC : projection (z.J) z.I,z.H = get_points(T.ABC : in_circle()) z.O = T.ABC.circumcenter C.OA = circle : new (z.O,z.A) T.IBA = triangle: new (z.I,z.B,z.A) z.w = T.IBA.circumcenter L.Ow = line : new (z.O,z.w) _,z.E = intersection (L.Ow, C.OA) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawArc(J,X)(Y) \tkzDrawCircles(I,H O,A) \tkzDrawCircle[red](w,I) \tkzDrawLines(Y,C A,B X,C E,w E,B) \tkzDrawSegments[blue](J,C J,K I,H I,O w,B) \tkzDrawPoints(A,B,C,I,J,E,w,H,K,O) \tkzLabelPoints(A,B,C,J,I,w,H,K,E,O) \tkzMarkRightAngles[fill=gray!20,opacity=.4](C,H,I A,K,J) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (5 , 0) z.C = point: new (-.4 , 4) T.ABC = triangle: new (z.A,z.B,z.C) z.J,z.K = get_points(T.ABC: ex_circle (2)) z.X ,z.Y,z.K = T.ABC : projection (z.J) z.I,z.H = get_points(T.ABC : in_circle()) z.O = T.ABC.circumcenter C.OA = circle : new (z.O,z.A) T.IBA = triangle: new (z.I,z.B,z.A) z.w = T.IBA.circumcenter L.Ow = line : new (z.O,z.w) _,z.E = intersection (L.Ow, C.OA) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawArc(J,X)(Y) \tkzDrawCircles(I,H O,A) \tkzDrawCircle[red](w,I) \tkzDrawLines(Y,C A,B X,C E,w E,B) \tkzDrawSegments[blue](J,C J,K I,H I,O w,B) \tkzDrawPoints(A,B,C,I,J,E,w,H,K,O) \tkzLabelPoints(A,B,C,J,I,w,H,K,E,O) \tkzMarkRightAngles[fill=gray!20,opacity=.4](C,H,I A,K,J) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection euler_relation (end) \subsection{External angle} % (fold) \label{sub:external_angle} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = .75 z.A = point: new (0 , 0) z.B = point: new (5 , 0) z.C = point: new (-2 , 4) T.ABC = triangle: new (z.A,z.B,z.C) T.ext = T.ABC: excentral () z.O = T.ABC.circumcenter z.D = intersection (T.ext.ab,T.ABC.ab) z.E = z.C: symmetry (z.B) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[purple,add=0 and .5](B,C) \tkzDrawSegment[purple](A,D) \tkzDrawSegment[orange](C,D) \tkzFillAngles[purple!30,opacity=.2](D,C,A E,C,D) \tkzMarkAngles[mark=|](D,C,A E,C,D) \tkzDrawPoints(A,...,D) \tkzLabelPoints[above](C) \tkzLabelPoints(A,B,D) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .75 z.A = point: new (0 , 0) z.B = point: new (5 , 0) z.C = point: new (-2 , 4) T.ABC = triangle: new (z.A,z.B,z.C) T.ext = T.ABC: excentral () z.O = T.ABC.circumcenter z.D = intersection (T.ext.ab,T.ABC.ab) z.E = z.C: symmetry (z.B) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[purple,add=0 and .5](B,C) \tkzDrawSegment[purple](A,D) \tkzDrawSegment[orange](C,D) \tkzFillAngles[purple!30,opacity=.2](D,C,A E,C,D) \tkzMarkAngles[mark=|](D,C,A E,C,D) \tkzDrawPoints(A,...,D) \tkzLabelPoints[above](C) \tkzLabelPoints(A,B,D) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection external_angle (end) \subsection{Internal angle} % (fold) \label{sub:internal_angle} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = .8 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (1 , 5) T = triangle: new (z.A,z.B,z.C) z.I = T.incenter L.AI = line: new (z.A,z.I) z.D = intersection (L.AI, T.bc) L.LL = T.ab: ll_from (z.C) L.AD = line: new (z.A,z.D) z.E = intersection (L.LL,L.AD) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[purple](C,E) \tkzDrawSegment[purple](A,E) \tkzFillAngles[purple!30,opacity=.4](B,A,C C,E,D) \tkzMarkAngles[mark=|](B,A,D D,A,C C,E,D) \tkzDrawPoints(A,...,E) \tkzLabelPoints(A,B) \tkzLabelPoints[above](C,D,E) \tkzMarkSegments(A,C C,E) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale =.8 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (1 , 5) T = triangle: new (z.A,z.B,z.C) z.I = T.incenter L.AI = line: new (z.A,z.I) z.D = intersection (L.AI, T.bc) L.LL = T.ab: ll_from (z.C) L.AD = line: new (z.A,z.D) z.E = intersection (L.LL,L.AD) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[purple](C,E) \tkzDrawSegment[purple](A,E) \tkzFillAngles[purple!30,opacity=.4](B,A,C C,E,D) \tkzMarkAngles[mark=|](B,A,D D,A,C C,E,D) \tkzDrawPoints(A,...,E) \tkzLabelPoints(A,B) \tkzLabelPoints[above](C,D,E) \tkzMarkSegments(A,C C,E) \end{tikzpicture} \end{minipage} %subsection internal_angle (end) \subsection{Feuerbach theorem} % (fold) \label{sub:nine_points} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = 1.5 z.A = point: new (0 , 0) z.B = point: new (5 , -.5) z.C = point: new (-.5 , 3) T.ABC = triangle: new (z.A,z.B,z.C) z.O = T.ABC.circumcenter z.N = T.ABC.eulercenter z.I,z.K = get_points(T.ABC: in_circle()) z.H = T.ABC.ab : projection (z.I) z.Ap, z.Bp, z.Cp = get_points (T.ABC : medial ()) C.IH = circle:new (z.I,z.H) C.NAp = circle:new (z.N,z.Ap) C.OA = circle:new (z.O,z.A) z.U = C.OA.south z.L = C.NAp.south z.M = C.NAp.north z.X = T.ABC.ab: projection (z.C) L.CU = line: new (z.C,z.U) L.ML = line: new (z.M,z.L) z.P = L.CU: projection (z.A) z.Q = L.CU: projection (z.B) L.LH = line: new (z.L,z.H) z.F = intersection (L.LH,C.IH) -- feuerbach \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawLine(L,F) \tkzDrawCircle[red](N,A') \tkzDrawCircle[blue](I,H) \tkzDrawCircles[teal](O,A L,C') \tkzDrawSegments(M,L B,U Q,C C,X A,P B,Q) \tkzDrawPolygons(A,B,C A',B',C') \tkzDrawPoints(A,B,C,N,H,A',B',C',U,L,M,P,Q,F,I) \tkzLabelPoints(A,B,C,N,H,A',B',C',U,L,M,P,Q,F,I) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = 1 z.A = point: new (0 , 0) z.B = point: new (5 , -.5) z.C = point: new (-.5 , 3) T.ABC = triangle: new (z.A,z.B,z.C) z.O = T.ABC.circumcenter z.N = T.ABC.eulercenter z.I,z.K = get_points(T.ABC: in_circle()) z.H = T.ABC.ab : projection (z.I) z.Ap,z.Bp,z.Cp = get_points (T.ABC : medial ()) C.IH = circle:new (z.I,z.H) C.NAp = circle:new (z.N,z.Ap) C.OA = circle:new (z.O,z.A) z.U = C.OA.south z.L = C.NAp.south z.M = C.NAp.north z.X = T.ABC.ab: projection (z.C) L.CU = line: new (z.C,z.U) L.ML = line: new (z.M,z.L) z.P = L.CU: projection (z.A) z.Q = L.CU: projection (z.B) L.LH = line: new (z.L,z.H) z.F = intersection (L.LH,C.IH) -- feuerbach \end{tkzelements} \hspace*{\fill} \begin{tikzpicture}[rotate=90] \tkzGetNodes \tkzDrawLine(L,F) \tkzDrawCircle[red](N,A') \tkzDrawCircle[blue](I,H) \tkzDrawCircles[teal](O,A L,C') \tkzDrawSegments(M,L B,U Q,C C,X A,P B,Q) \tkzDrawPolygons(A,B,C A',B',C') \tkzDrawPoints(A,B,C,N,H,A',B',C',U,L,M,P,Q,F,I) \tkzLabelPoints(A,B,C,N,H,A',B',C',U,L,M,P,Q,F,I) \end{tikzpicture} \end{minipage} % subsection nine_points (end) \subsection{Gold ratio with segment} % (fold) \label{sub:gold_ratio_with_segment} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (8 , 0) L.AB = line: new (z.A,z.B) _,_,z.X,z.Y = get_points(L.AB: square ()) L.BX = line: new (z.B,z.X) z.M = L.BX.mid C.MA = circle: new (z.M,z.A) _,z.K = intersection (L.BX,C.MA) L.AK = line: new (z.Y,z.K) z.C = intersection (L.AK,L.AB) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines(A,B X,K) \tkzDrawLine[teal](Y,K) \tkzDrawPoints(A,B,C,X,Y,M,K) \tkzDrawArc[delta=20](M,A)(K) \tkzLabelPoints(A,B,C) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} scale = .5 z.A = point: new (0 , 0) z.B = point: new (8 , 0) L.AB = line: new (z.A,z.B) _,_,z.X,z.Y = get_points(L.AB: square ()) L.BX = line: new (z.B,z.X) z.M = L.BX.mid C.MA = circle: new (z.M,z.A) _,z.K = intersection (L.BX,C.MA) L.AK = line: new (z.Y,z.K) z.C = intersection (L.AK,L.AB) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines(A,B X,K) \tkzDrawLine[teal](Y,K) \tkzDrawPoints(A,B,C,X,Y,M,K) \tkzDrawArc[delta=20](M,A)(K) \tkzLabelPoints(A,B,C) \end{tikzpicture} \end{minipage} %subsection gold_ratio_with_segment (end) \subsection{Gold Arbelos} % (fold) \label{sub:gold_arbelos} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} scale = .6 z.A = point: new (0 , 0) z.C = point: new (6 , 0) L.AC = line: new (z.A,z.C) _,_,z.x,z.y = get_points(L.AC: square ()) z.O_1 = L.AC . mid C = circle: new (z.O_1,z.x) z.B = intersection (L.AC,C) L.CB = line: new (z.C,z.B) z.O_2 = L.CB.mid L.AB = line: new (z.A,z.B) z.O_0 = L.AB.mid \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O_1,C O_2,B O_0,B) \tkzDrawPoints(A,C,B,O_1,O_2,O_0) \tkzLabelPoints(A,C,B) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} scale = .4 z.A = point: new (0 , 0) z.C = point: new (6 , 0) L.AC = line: new (z.A,z.C) _,_,z.x,z.y = get_points(L.AC: square ()) z.O_1 = L.AC . mid C = circle: new (z.O_1,z.x) z.B = intersection (L.AC,C) L.CB = line: new (z.C,z.B) z.O_2 = L.CB.mid L.AB = line: new (z.A,z.B) z.O_0 = L.AB.mid \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O_1,C O_2,B O_0,B O_1,x) \tkzDrawPoints(A,C,B,O_1,O_2,O_0,x) \tkzLabelPoints[below right](A,C,B) \end{tikzpicture} \end{minipage} % subsection gold_arbelos (end) \subsection{Harmonic division v1} % (fold) \label{sub:harmonic_division_v1} \begin{minipage}[t]{.4\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale=.75 z.A = point: new (0 , 0) z.B = point: new (4 , 0) z.G = point: new (2,2) L.AG = line : new (z.A,z.G) L.AB = line : new (z.A,z.B) z.E = L.AG : colinear_at (z.B,.5) L.GE = line : new (z.G,z.E) z.D = intersection (L.GE,L.AB) z.F = z.B : symmetry (z.E) L.GF = line :new (z.G,z.F) z.C = intersection (L.GF,L.AB) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines(A,B A,G A,D A,G F,E G,F G,D) \tkzDrawPoints(A,B,G,E,F,C,D) \tkzLabelPoints(A,B,G,E,F,C,D) \tkzMarkSegments(F,B B,E) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.6\textwidth}\vspace{0pt}% \begin{tkzelements} scale=.75 z.A = point: new (0 , 0) z.B = point: new (4 , 0) z.G = point: new (2,2) L.AG = line : new (z.A,z.G) L.AB = line : new (z.A,z.B) z.E = L.AG : colinear_at (z.B,.5) L.GE = line : new (z.G,z.E) z.D = intersection (L.GE,L.AB) z.F = z.B : symmetry (z.E) L.GF = line :new (z.G,z.F) z.C = intersection (L.GF,L.AB) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines(A,B A,G A,D A,G F,E G,F G,D) \tkzDrawPoints(A,B,G,E,F,C,D) \tkzLabelPoints(A,B,G,E,F,C,D) \tkzMarkSegments(F,B B,E) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection harmonic_division_v1 (end) \subsection{Harmonic division v2} % (fold) \label{sub:harmonic_division_v2} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = .5 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.D = point: new (12 , 0) L.AB = line: new (z.A,z.B) z.X = L.AB.north_pa L.XB = line: new (z.X,z.B) z.E = L.XB.mid L.ED = line: new (z.E,z.D) L.AX = line: new (z.A,z.X) L.AE = line: new (z.A,z.E) z.F = intersection (L.ED,L.AX) L.BF = line: new (z.B,z.F) z.G = intersection (L.AE,L.BF) L.GX = line: new (z.G,z.X) z.C = intersection (L.GX,L.AB) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines(A,D A,E B,F D,F X,A X,B X,C) \tkzDrawPoints(A,...,G,X) \tkzLabelPoints(A,...,G,X) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .5 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.D = point: new (12 , 0) L.AB = line: new (z.A,z.B) z.X = L.AB.north_pa L.XB = line: new (z.X,z.B) z.E = L.XB.mid L.ED = line: new (z.E,z.D) L.AX = line: new (z.A,z.X) L.AE = line: new (z.A,z.E) z.F = intersection (L.ED,L.AX) L.BF = line: new (z.B,z.F) z.G = intersection (L.AE,L.BF) L.GX = line: new (z.G,z.X) z.C = intersection (L.GX,L.AB) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines(A,D A,E B,F D,F X,A X,B X,C) \tkzDrawPoints(A,...,G,X) \tkzLabelPoints(A,...,G,X) \end{tikzpicture} \end{minipage} % subsection harmonic_division_v2 (end) \subsection{Menelaus} % (fold) \label{sub:menelaus} \begin{minipage}{.4\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (5 , 4) z.P = point: new (-1 , 0) z.X = point: new (6 , 3) L.AC = line: new (z.A,z.C) L.PX = line: new (z.P,z.X) L.BC = line: new (z.B,z.C) z.Q = intersection (L.AC,L.PX) z.R = intersection (L.BC,L.PX) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[new](P,R) \tkzDrawLines(P,B A,C B,C) \tkzDrawPoints(P,Q,R,A,B,C) \tkzLabelPoints(A,B,C,P,Q,R) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (5 , 4) z.P = point: new (-1 , 0) z.X = point: new (6 , 3) L.AC = line: new (z.A,z.C) L.PX = line: new (z.P,z.X) L.BC = line: new (z.B,z.C) z.Q = intersection (L.AC,L.PX) z.R = intersection (L.BC,L.PX) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawLine[new](P,R) \tkzDrawLines(P,B A,C B,C) \tkzDrawPoints(P,Q,R,A,B,C) \tkzLabelPoints(A,B,C,P,Q,R) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection menelaus (end) \subsection{Radical axis v1} % (fold) \label{sub:radical_axis_v1} \begin{Verbatim} \begin{tkzelements} scale = .75 z.X = point : new (0,0) z.B = point : new (2,2) z.Y = point : new (7,1) z.Ap = point : new (8,-1) L.XY = line : new (z.X,z.Y) C.XB = circle : new (z.X,z.B) C.YAp = circle : new (z.Y,z.Ap) z.E,z.F = get_points (C.XB : radical_axis (C.YAp)) z.A = C.XB : point (0.4) T.ABAp = triangle: new (z.A,z.B,z.Ap) z.O = T.ABAp.circumcenter C.OAp = circle : new (z.O,z.Ap) _,z.Bp = intersection (C.OAp,C.YAp) L.AB = line : new (z.A,z.B) L.ApBp = line : new (z.Ap,z.Bp) z.M = intersection (L.AB,L.ApBp) z.H = L.XY : projection (z.M) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(X,B Y,A') \tkzDrawArc[dashed,delta=30](O,A')(A) \tkzDrawPoints(A,B,A',B',M,H,X,Y,O,E,F) \tkzDrawLines[red](A,M A',M X,Y E,F) \tkzDrawLines[red,add=1 and 3](M,H) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = .75 z.X = point : new (0,0) z.B = point : new (2,2) z.Y = point : new (7,1) z.Ap = point : new (8,-1) L.XY = line : new (z.X,z.Y) C.XB = circle : new (z.X,z.B) C.YAp = circle : new (z.Y,z.Ap) z.E,z.F = get_points (C.XB : radical_axis (C.YAp)) z.A = C.XB : point (0.4) T.ABAp = triangle: new (z.A,z.B,z.Ap) z.O = T.ABAp.circumcenter C.OAp = circle : new (z.O,z.Ap) _,z.Bp = intersection (C.OAp,C.YAp) L.AB = line : new (z.A,z.B) L.ApBp = line : new (z.Ap,z.Bp) z.M = intersection (L.AB,L.ApBp) z.H = L.XY : projection (z.M) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(X,B Y,A') \tkzDrawArc[dashed,delta=30](O,A')(A) \tkzDrawPoints(A,B,A',B',M,H,X,Y,O,E,F) \tkzDrawLines[red](A,M A',M X,Y E,F) \tkzDrawLines[red,add=1 and 3](M,H) \end{tikzpicture} % subsection radical_axis_v1 (end) \subsection{Radical axis v2} % (fold) \label{sub:radical_axis_v2} \begin{Verbatim} \begin{tkzelements} scale = 1.25 z.O = point : new (-1,0) z.Op = point : new (4,-1) z.B = point : new (0,2) z.D = point : new (4,0) C.OB = circle : new (z.O,z.B) C.OpD = circle : new (z.Op,z.D) L.EF = C.OB : radical_axis (C.OpD) z.E,z.F = get_points (L.EF) z.M = L.EF : point (.75) L.MT,L.MTp = C.OB : tangent_from (z.M) _,z.T = get_points (L.MT) _,z.Tp = get_points (L.MTp) L.MK,L.MKp = C.OpD : tangent_from (z.M) _,z.K = get_points (L.MK) _,z.Kp = get_points (L.MKp) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',D) \tkzDrawLine(E,F) \tkzDrawLine[add=.5 and .5](O,O') \tkzDrawLines[add = 0 and .5](M,T M,T' M,K M,K') \tkzDrawCircle(M,T) \tkzDrawPoints(O,O',T,M,T',K,K') \tkzLabelPoints(O,O',T,T',K,K',M) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale =1.25 z.O = point : new (-1,0) z.Op = point : new (4,-1) z.B = point : new (0,2) z.D = point : new (4,0) C.OB = circle : new (z.O,z.B) C.OpD = circle : new (z.Op,z.D) L.EF = C.OB : radical_axis (C.OpD) z.E,z.F = get_points (L.EF) z.M = L.EF : point (.75) L.MT,L.MTp = C.OB : tangent_from (z.M) _,z.T = get_points (L.MT) _,z.Tp = get_points (L.MTp) L.MK,L.MKp = C.OpD : tangent_from (z.M) _,z.K = get_points (L.MK) _,z.Kp = get_points (L.MKp) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',D) \tkzDrawLine(E,F) \tkzDrawLine[add=.5 and .5](O,O') \tkzDrawLines[add = 0 and .5](M,T M,T' M,K M,K') \tkzDrawCircle(M,T) \tkzDrawPoints(O,O',T,M,T',K,K') \tkzLabelPoints(O,O',T,T',K,K',M) \end{tikzpicture} \hspace*{\fill} % subsection radical_axis_v2 (end) \subsection{Radical axis v3} % (fold) \label{sub:radical_axis_v3} \begin{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.B = point : new (4,0) z.Op = point : new (6,0) C.OB = circle : new (z.O,z.B) C.OpB = circle : new (z.Op,z.B) L.EF = C.OB : radical_axis (C.OpB) z.E,z.F = get_points(L.EF) z.M = L.EF : point (0.2) L = C.OB : tangent_from (z.M) _,z.T = get_points (L) L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',B) \tkzDrawSegments(M,T M,T') \tkzDrawSegments(E,F) \tkzDrawLine[add=.5 and .5](O,O') \tkzDrawPoints(O,B,O',E,F,M,T,T') \tkzLabelPoints(O,O',B,E,F,T,T') \tkzDrawArc(M,T')(T) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.B = point : new (4,0) z.Op = point : new (6,0) C.OB = circle : new (z.O,z.B) C.OpB = circle : new (z.Op,z.B) L.EF = C.OB : radical_axis (C.OpB) z.E,z.F = get_points(L.EF) z.M = L.EF : point (0.2) L = C.OB : tangent_from (z.M) _,z.T = get_points (L) L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',B) \tkzDrawSegments(M,T M,T') \tkzDrawSegments(E,F) \tkzDrawLine[add=.5 and .5](O,O') \tkzDrawPoints(O,B,O',E,F,M,T,T') \tkzLabelPoints(O,O',B,E,F,T,T') \tkzDrawArc(M,T')(T) \end{tikzpicture} \hspace*{\fill} % subsection radical_axis_v3 (end) \subsection{Radical axis v4} % (fold) \label{sub:radical_axis_v4} \begin{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.B = point : new (5,0) z.Op = point : new (3,0) C.OB = circle : new (z.O,z.B) C.OpB = circle : new (z.Op,z.B) L.EF = C.OB : radical_axis (C.OpB) z.E,z.F = get_points(L.EF) z.H = L.EF.mid z.M = L.EF : point (.8) _,L = C.OB : tangent_from (z.M) _,z.T = get_points (L) _,L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',B) \tkzDrawSegments(M,T M,T') \tkzDrawSegments(E,F) \tkzDrawLine[add=.3 and .3](O,H) \tkzDrawPoints(O,O',B,E,H,M) \tkzLabelPoints[below right](O,O',E,F,M,T,T') \tkzDrawArc(M,B)(T) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.B = point : new (5,0) z.Op = point : new (3,0) C.OB = circle : new (z.O,z.B) C.OpB = circle : new (z.Op,z.B) L.EF = C.OB : radical_axis (C.OpB) z.E,z.F = get_points(L.EF) z.H = L.EF.mid z.M = L.EF : point (.8) _,L = C.OB : tangent_from (z.M) _,z.T = get_points (L) _,L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',B) \tkzDrawSegments(M,T M,T') \tkzDrawSegments(E,F) \tkzDrawLine[add=.3 and .3](O,H) \tkzDrawPoints(O,O',B,E,H,M) \tkzLabelPoints[below right](O,O',E,F,M,T,T') \tkzDrawArc(M,B)(T) \end{tikzpicture} \hspace*{\fill} % subsection radical_axis_v4 (end) \subsection{Radical center} % (fold) \label{sub:radical_center} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.x = point : new (1,0) z.y = point : new (4,0) z.z = point : new (2,0) z.Op = point : new (4,2) z.P = point : new (2,2.5) C.Ox = circle : new (z.O,z.x) C.Pz = circle : new (z.P,z.z) C.Opy = circle : new (z.Op,z.y) z.ap,z.a = intersection (C.Ox,C.Pz) z.bp,z.b = intersection (C.Opy,C.Pz) L.aap = line : new (z.a,z.ap) L.bbp = line : new (z.b,z.bp) z.X = intersection (L.aap,L.bbp) -- or z.X = radical_center(C.Ox,C.Pz,C.Opy) L.OOp = line : new (z.O,z.Op) z.H = L.OOp : projection (z.X) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,a O',b P,z) \tkzDrawLines[red](a,X b',X H,X O,O') \tkzDrawPoints(O,O',P,a,a',b,b',X,H) \tkzLabelPoints[below right](O,O',P,H) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} z.O = point : new (0,0) z.x = point : new (1,0) z.y = point : new (4,0) z.z = point : new (2,0) z.Op = point : new (4,2) z.P = point : new (2,2.5) C.Ox = circle : new (z.O,z.x) C.Pz = circle : new (z.P,z.z) C.Opy = circle : new (z.Op,z.y) z.ap,z.a = intersection (C.Ox,C.Pz) z.bp,z.b = intersection (C.Opy,C.Pz) L.aap = line : new (z.a,z.ap) L.bbp = line : new (z.b,z.bp) z.X = intersection (L.aap,L.bbp) L.OOp = line : new (z.O,z.Op) z.H = L.OOp : projection (z.X) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,a O',b P,z) \tkzDrawLines[red](a,X b',X H,X O,O') \tkzDrawPoints(O,O',P,a,a',b,b',X,H) \tkzLabelPoints[below right](O,O',P,H) \end{tikzpicture} \end{minipage} % subsection radical_center (end) \subsection{Radical circle} % (fold) \label{sub:radical_circle} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = .5 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (0.8,4) T.ABC = triangle : new ( z.A,z.B,z.C ) C.exa = T.ABC : ex_circle () z.I_a,z.Xa = get_points (C.exa) C.exb = T.ABC : ex_circle (1) z.I_b,z.Xb = get_points (C.exb) C.exc = T.ABC : ex_circle (2) z.I_c,z.Xc = get_points (C.exc) C.ortho = C.exa : radical_circle (C.exb,C.exc) z.w,z.a = get_points (C.ortho) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawCircles(I_a,Xa I_b,Xb I_c,Xc) \tkzDrawCircles[red,thick](w,a) \tkzDrawPoints(A,B,C) \tkzLabelPoints(A,B,C) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{tkzelements} scale = .5 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (0.8,4) T.ABC = triangle : new ( z.A,z.B,z.C ) C.exa = T.ABC : ex_circle () z.I_a,z.Xa = get_points (C.exa) C.exb = T.ABC : ex_circle (1) z.I_b,z.Xb = get_points (C.exb) C.exc = T.ABC : ex_circle (2) z.I_c,z.Xc = get_points (C.exc) C.ortho = C.exa : radical_circle (C.exb,C.exc) z.w,z.a = get_points (C.ortho) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawCircles(I_a,Xa I_b,Xb I_c,Xc) \tkzDrawCircles[red,thick](w,a) \tkzDrawPoints(A,B,C) \tkzLabelPoints(A,B,C) \end{tikzpicture} \hspace*{\fill} % subsection radical_circle (end) \subsection{Euler ellipse} % (fold) \label{sub:hexagram} \begin{tkzelements} scale =1.3 z.A = point: new (0 , 0) z.B = point: new (5 , 1) L.AB = line : new (z.A,z.B) z.C = point: new (.8 , 3) T.ABC = triangle: new (z.A,z.B,z.C) z.N = T.ABC.eulercenter z.G = T.ABC.centroid z.O = T.ABC.circumcenter z.H = T.ABC.orthocenter z.Ma,z.Mb,z.Mc = get_points (T.ABC : medial ()) z.Ha,z.Hb,z.Hc = get_points (T.ABC : orthic ()) z.Ea,z.Eb,z.Ec = get_points (T.ABC: extouch()) L.euler = T.ABC : euler_line () C.circum = T.ABC : circum_circle () C.euler = T.ABC : euler_circle () z.I,z.J = intersection (L.euler,C.euler) E = ellipse: foci (z.H,z.O,z.I) a = E.Rx b = E.Ry ang = math.deg(E.slope) L.AH = line: new (z.A,z.H) L.BH = line: new (z.B,z.H) L.CH = line: new (z.C,z.H) z.X = intersection (L.AH,C.circum) _,z.Y = intersection (L.BH,C.circum) _,z.Z = intersection (L.CH,C.circum) L.BC = line: new (z.B,z.C) L.XO = line: new (z.X,z.O) L.YO = line: new (z.Y,z.O) L.ZO = line: new (z.Z,z.O) z.x = intersection (L.BC,L.XO) z.U = intersection (L.XO,E) _,z.V = intersection (L.YO,E) _,z.W = intersection (L.ZO,E) \end{tkzelements} \begin{minipage}{.4\textwidth} \begin{Verbatim} \begin{tkzelements} scale = 1.3 z.A = point: new (0 , 0) z.B = point: new (5 , 1) L.AB = line : new (z.A,z.B) z.C = point: new (.8 , 3) T.ABC = triangle: new (z.A,z.B,z.C) z.N = T.ABC.eulercenter z.G = T.ABC.centroid z.O = T.ABC.circumcenter z.H = T.ABC.orthocenter z.Ma,z.Mb, z.Mc = get_points(T.ABC:medial ()) z.Ha,z.Hb, z.Hc = get_points(T.ABC:orthic ()) z.Ea,z.Eb, z.Ec = get_points(T.ABC:extouch()) L.euler = T.ABC : euler_line () C.circum = T.ABC : circum_circle () C.euler = T.ABC : euler_circle () z.I,z.J = intersection (L.euler,C.euler) E = ellipse: foci (z.H,z.O,z.I) a = E.Rx b = E.Ry ang = math.deg(E.slope) L.AH = line: new (z.A,z.H) L.BH = line: new (z.B,z.H) L.CH = line: new (z.C,z.H) z.X = intersection (L.AH,C.circum) _,z.Y = intersection (L.BH,C.circum) _,z.Z = intersection (L.CH,C.circum) L.BC = line: new (z.B,z.C) L.XO = line: new (z.X,z.O) L.YO = line: new (z.Y,z.O) L.ZO = line: new (z.Z,z.O) z.x = intersection (L.BC,L.XO) z.U = intersection (L.XO,E) _,z.V = intersection (L.YO,E) _,z.W = intersection (L.ZO,E) \end{tkzelements} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawCircles[red](N,Ma O,A) \tkzDrawSegments(A,X B,Y C,Z B,Hb C,Hc X,O Y,O Z,O) \tkzDrawPolygon[red](U,V,W) \tkzLabelPoints[red](U,V,W) \tkzLabelPoints(A,B,C,X,Y,Z) \tkzDrawLine[blue](I,J) \tkzLabelPoints[blue,right](O,N,G,H,I,J) \tkzDrawPoints(I,J,U,V,W) \tkzDrawPoints(A,B,C,N,G,H,O,X,Y,Z,Ma,Mb,Mc,Ha,Hb,Hc) \tkzDrawEllipse[blue](N,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{ang}) \end{tikzpicture} \end{minipage} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawCircles[red](N,Ma O,A) \tkzDrawSegments(A,X B,Y C,Z B,Hb C,Hc X,O Y,O Z,O) \tkzDrawPolygon[red](U,V,W) \tkzLabelPoints[red](U,V,W) \tkzLabelPoints(A,B,C,X,Y,Z) \tkzDrawLine[blue](I,J) \tkzLabelPoints[blue,right](O,N,G,H,I,J) \tkzDrawPoints(I,J,U,V,W) \tkzDrawPoints(A,B,C,N,G,H,O,X,Y,Z,Ma,Mb,Mc,Ha,Hb,Hc) \tkzDrawEllipse[blue](N,\tkzUseLua{a},\tkzUseLua{b},\tkzUseLua{ang}) \end{tikzpicture} \end{Verbatim} % subsection hexagram (end) \subsection{Gold Arbelos properties} % (fold) \label{sub:gold_arbelos_properties} \begin{tkzelements} scale = .7 z.A = point : new(0,0) z.B = point : new(10,0) z.C = gold_segment_ (z.A,z.B) L.AB = line:new (z.A,z.B) z.O_1 = L.AB.mid L.AC = line:new (z.A,z.C) z.O_2 = L.AC.mid L.CB = line:new (z.C,z.B) z.O_3 = L.CB.mid C1 = circle:new (z.O_1,z.B) C2 = circle:new (z.O_2,z.C) C3 = circle:new (z.O_3,z.B) z.Q = C2.north z.P = C3.north L1 = line:new (z.O_2,z.O_3) z.M_0 = L1:harmonic_ext (z.C) L2 = line:new (z.O_1,z.O_2) z.M_1 = L2:harmonic_int (z.A) L3 = line:new (z.O_1,z.O_3) z.M_2 = L3:harmonic_int (z.B) Lbq = line:new (z.B,z.Q) Lap = line:new (z.A,z.P) z.S = intersection (Lbq,Lap) z.x = z.C: north () L = line : new (z.C,z.x) z.D,_ = intersection (L,C1) L.CD = line :new (z.C,z.D) z.O_7 = L.CD.mid C.DC = circle: new (z.D,z.C) z.V,z.U = intersection (C.DC,C1) L.UV = line :new (z.U,z.V) z.R ,z.S = L.UV : projection (z.O_2,z.O_3) L.O1D = line : new (z.O_1,z.D) z.W = intersection (L.UV,L.O1D) z.O = C.DC : inversion (z.W) \end{tkzelements} \begin{minipage}{.4\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point : new(0,0) z.B = point : new(10,0) z.C = gold_segment_ (z.A,z.B) L.AB = line:new (z.A,z.B) z.O_1 = L.AB.mid L.AC = line:new (z.A,z.C) z.O_2 = L.AC.mid L.CB = line:new (z.C,z.B) z.O_3 = L.CB.mid C1 = circle:new (z.O_1,z.B) C2 = circle:new (z.O_2,z.C) C3 = circle:new (z.O_3,z.B) z.Q = C2.north z.P = C3.north L1 = line:new (z.O_2,z.O_3) z.M_0 = L1:harmonic_ext (z.C) L2 = line:new (z.O_1,z.O_2) z.M_1 = L2:harmonic_int (z.A) L3 = line:new (z.O_1,z.O_3) z.M_2 = L3:harmonic_int (z.B) Lbq = line:new (z.B,z.Q) Lap = line:new (z.A,z.P) z.S = intersection (Lbq,Lap) z.x = z.C: north () L = line : new (z.C,z.x) z.D,_ = intersection (L,C1) L.CD = line :new (z.C,z.D) z.O_7 = L.CD.mid C.DC = circle: new (z.D,z.C) z.U,z.V = intersection (C.DC,C1) L.UV = line :new (z.U,z.V) z.R ,z.S = L.UV : projection (z.O_2,z.O_3) L.O1D = line : new (z.O_1,z.D) z.W = intersection (L.UV,L.O1D) z.O = C.DC : inversion (z.W) \end{tkzelements} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal](O_1,B) \tkzDrawSemiCircles[thin,teal](O_2,C O_3,B) \tkzDrawArc[purple,delta=0](D,V)(U) \tkzDrawCircle[new](O_7,C) \tkzDrawSegments[thin,purple](A,D D,B C,R C,S C,D U,V) \tkzDrawSegments[thin,red](O,D A,O O,B) \tkzDrawPoints(A,B,C,D,O_7) %, \tkzDrawPoints(O_1,O_2,O_3,U,V,R,S,W,O) \tkzDrawSegments[cyan](O_3,S O_2,R) \tkzDrawSegments[very thin](A,B) \tkzDrawSegments[cyan,thin](C,U U,D) \tkzMarkRightAngles[size=.2,fill=gray!40,opacity=.4](D,C,A A,D,B D,S,C D,W,V O_3,S,U O_2,R,U) \tkzFillAngles[cyan!40,opacity=.4](B,A,D A,D,O_1 C,D,B D,C,R B,C,S A,R,O_2) \tkzFillAngles[green!40,opacity=.4](S,C,D W,R,D D,B,C R,C,A O_3,S,B) \tkzLabelPoints[below](C,O_2,O_3,O_1) \tkzLabelPoints[above](D) \tkzLabelPoints[below](O) \tkzLabelPoints[below left](A) \tkzLabelPoints[above left](R) \tkzLabelPoints[above right](S) \tkzLabelPoints[left](V) \tkzLabelPoints[below right](B,U,W,O_7) \end{tikzpicture} \hspace*{\fill} \end{minipage} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal](O_1,B) \tkzDrawSemiCircles[thin,teal](O_2,C O_3,B) \tkzDrawArc[purple,delta=0](D,V)(U) \tkzDrawCircle[new](O_7,C) \tkzDrawSegments[thin,purple](A,D D,B C,R C,S C,D U,V) \tkzDrawSegments[thin,red](O,D A,O O,B) \tkzDrawPoints(A,B,C,D,O_7) %, \tkzDrawPoints(O_1,O_2,O_3,U,V,R,S,W,O) \tkzDrawSegments[cyan](O_3,S O_2,R) \tkzDrawSegments[very thin](A,B) \tkzDrawSegments[cyan,thin](C,U U,D) \tkzMarkRightAngles[size=.2,fill=gray!40,opacity=.4](D,C,A A,D,B D,S,C D,W,V O_3,S,U O_2,R,U) \tkzFillAngles[cyan!40,opacity=.4](B,A,D A,D,O_1 C,D,B D,C,R B,C,S A,R,O_2) \tkzFillAngles[green!40,opacity=.4](S,C,D W,R,D D,B,C R,C,A O_3,S,B) \tkzLabelPoints[below](C,O_2,O_3,O_1) \tkzLabelPoints[above](D) \tkzLabelPoints[below](O) \tkzLabelPoints[below left](A) \tkzLabelPoints[above left](R) \tkzLabelPoints[above right](S) \tkzLabelPoints[left](V) \tkzLabelPoints[below right](B,U,W,O_7) \end{tikzpicture} \end{Verbatim} % subsection gold_arbelos_properties (end) \subsection{Apollonius circle v1 with inversion} % (fold) \label{sub:apollonius_circle_v1_with_inversion} \begin{Verbatim} \begin{tkzelements} scale = .7 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (0.8,4) T.ABC = triangle : new ( z.A,z.B,z.C ) z.N = T.ABC.eulercenter z.Ea,z.Eb,z.Ec = get_points ( T.ABC : feuerbach () ) z.Ja,z.Jb,z.Jc = get_points ( T.ABC : excentral () ) z.S = T.ABC : spieker_center () C.JaEa = circle : new (z.Ja,z.Ea) C.ortho = circle : radius (z.S,math.sqrt(C.JaEa : power (z.S) )) z.a = C.ortho.south C.euler = T.ABC: euler_circle () C.apo = C.ortho : inversion (C.euler) z.O = C.apo.center z.xa,z.xb,z.xc = C.ortho : inversion (z.Ea,z.Eb,z.Ec) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[red](O,xa N,Ea) \tkzFillCircles[green!30!black,opacity=.3](O,xa) \tkzFillCircles[yellow!30,opacity=.7](Ja,Ea Jb,Eb Jc,Ec) \tkzFillCircles[teal!30!black,opacity=.3](S,a) \tkzFillCircles[green!30,opacity=.3](N,Ea) \tkzDrawPoints[red](Ea,Eb,Ec,xa,xb,xc,N) \tkzClipCircle(O,xa) \tkzDrawLines[add=3 and 3](A,B A,C B,C) \tkzDrawCircles(Ja,Ea Jb,Eb Jc,Ec) \tkzFillCircles[lightgray!30,opacity=.7](Ja,Ea Jb,Eb Jc,Ec) \tkzDrawCircles[teal](S,a) \tkzDrawPoints(A,B,C,O) \tkzDrawPoints[teal](S) \tkzLabelPoints(A,B,C,O,S,N) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = .5 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (0.8,4) T.ABC = triangle : new ( z.A,z.B,z.C ) z.N = T.ABC.eulercenter z.Ea,z.Eb,z.Ec = get_points ( T.ABC : feuerbach () ) z.Ja,z.Jb,z.Jc = get_points ( T.ABC : excentral () ) z.S = T.ABC : spieker_center () C.JaEa = circle : new (z.Ja,z.Ea) C.ortho = circle : radius (z.S,math.sqrt(C.JaEa : power (z.S) )) z.a = C.ortho.south C.euler = T.ABC: euler_circle () C.apo = C.ortho : inversion (C.euler) z.O = C.apo.center z.xa,z.xb,z.xc = C.ortho : inversion (z.Ea,z.Eb,z.Ec) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[red](O,xa N,Ea) \tkzFillCircles[green!30!black,opacity=.3](O,xa) \tkzFillCircles[yellow!30,opacity=.7](Ja,Ea Jb,Eb Jc,Ec) \tkzFillCircles[teal!30!black,opacity=.3](S,a) \tkzFillCircles[green!30,opacity=.3](N,Ea) \tkzDrawPoints[red](Ea,Eb,Ec,xa,xb,xc,N) \tkzClipCircle(O,xa) \tkzDrawLines[add=3 and 3](A,B A,C B,C) \tkzDrawCircles(Ja,Ea Jb,Eb Jc,Ec) \tkzFillCircles[lightgray!30,opacity=.7](Ja,Ea Jb,Eb Jc,Ec) \tkzDrawCircles[teal](S,a) \tkzDrawPoints(A,B,C,O) \tkzDrawPoints[teal](S) \tkzLabelPoints(A,B,C,O,S,N) \end{tikzpicture} \hspace*{\fill} % subsection apollonius_circle_v1_with_inversion (end) \subsection{Apollonius circle v2} % (fold) \label{sub:apollonius_circle_v2} \begin{Verbatim} \begin{tkzelements} scale = .5 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (0.8,4) T.ABC = triangle: new(z.A,z.B,z.C) z.O = T.ABC.circumcenter z.H = T.ABC.orthocenter z.G = T.ABC.centroid z.L = T.ABC: lemoine_point () z.S = T.ABC: spieker_center () C.euler = T.ABC: euler_circle () z.N,z.Ma = get_points (C.euler) C.exA = T.ABC : ex_circle () z.Ja,z.Xa = get_points (C.exA) C.exB = T.ABC : ex_circle (1) z.Jb,z.Xb = get_points (C.exB) C.exC = T.ABC : ex_circle (2) z.Jc,z.Xc = get_points (C.exC) L.OL = line: new (z.O,z.L) L.NS = line: new (z.N,z.S) z.o = intersection (L.OL,L.NS) -- center of Apollonius circle L.NMa = line: new (z.N,z.Ma) L.ox = L.NMa: ll_from (z.o) L.MaS = line: new (z.Ma,z.S) z.t = intersection (L.ox,L.MaS) -- through \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines[add=1 and 1](A,B A,C B,C) \tkzDrawCircles(Ja,Xa Jb,Xb Jc,Xc o,t N,Ma) % \tkzClipCircle(o,t) \tkzDrawLines[red](o,L N,o Ma,t) \tkzDrawLines[cyan,add=4 and 4](Ma,N o,t) \tkzDrawPoints(A,B,C,Ma,Ja,Jb,Jc) \tkzDrawPoints[red](N,O,L,S,o,t) \tkzLabelPoints[right,font=\tiny](A,B,C,Ja,Jb,Jc,O,N,L,S,Ma,o) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale =.5 z.A = point: new (0,0) z.B = point: new (6,0) z.C = point: new (0.8,4) T.ABC = triangle: new(z.A,z.B,z.C) z.O = T.ABC.circumcenter z.H = T.ABC.orthocenter z.G = T.ABC.centroid z.L = T.ABC: lemoine_point () z.S = T.ABC: spieker_center () C.euler = T.ABC: euler_circle () z.N,z.Ma = get_points (C.euler) C.exA = T.ABC : ex_circle () z.Ja,z.Xa = get_points (C.exA) C.exB = T.ABC : ex_circle (1) z.Jb,z.Xb = get_points (C.exB) C.exC = T.ABC : ex_circle (2) z.Jc,z.Xc = get_points (C.exC) L.OL = line: new (z.O,z.L) L.NS = line: new (z.N,z.S) z.o = intersection (L.OL,L.NS) -- center of Apollonius circle L.NMa = line: new (z.N,z.Ma) L.ox = L.NMa: ll_from (z.o) L.MaS = line: new (z.Ma,z.S) z.t = intersection (L.ox,L.MaS) -- through \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawLines[add=1 and 1](A,B A,C B,C) \tkzDrawCircles(Ja,Xa Jb,Xb Jc,Xc o,t N,Ma) % \tkzClipCircle(o,t) \tkzDrawLines[red](o,L N,o Ma,t) \tkzDrawLines[cyan,add=4 and 4](Ma,N o,t) \tkzDrawPoints(A,B,C,Ma,Ja,Jb,Jc) \tkzDrawPoints[red](N,O,L,S,o,t) \tkzLabelPoints[right,font=\tiny](A,B,C,Ja,Jb,Jc,O,N,L,S,Ma,o) \end{tikzpicture} \hspace*{\fill} % subsection apollonius_circle_v2 (end) \subsection{Orthogonal circles v1} % (fold) \label{sub:orthogonal_circles_v1} \begin{Verbatim} \begin{tkzelements} scale = .6 z.C_1 = point: new (0,0) z.C_2 = point: new (8,0) z.A = point: new (5,0) C = circle: new (z.C_1,z.A) z.S,z.T = get_points (C: orthogonal_from (z.C_2)) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_1,T C_2,T) \tkzDrawSegments(C_1,T C_2,T) \tkzDrawLine(C_1,C_2) \tkzMarkRightAngle[fill=teal,% opacity=.2,size=1](C_1,T,C_2) \tkzDrawPoints(C_1,C_2,T) \tkzLabelPoints(C_1,C_2) \tkzLabelPoints[above](T) \tkzLabelSegment[left](C_1,T){r} \tkzLabelSegments[right](C_2,T){$\gamma$} \tkzLabelSegment[below](C_1,C_2){d} \tkzLabelCircle[left=10pt](C_1,T)(180){Circle 1} \tkzLabelCircle[right=10pt](C_2,T)(180){Circle 2} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = .6 z.C_1 = point: new (0,0) z.C_2 = point: new (8,0) z.A = point: new (5,0) C = circle: new (z.C_1,z.A) z.S,z.T = get_points (C: orthogonal_from (z.C_2)) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_1,T C_2,T) \tkzDrawSegments(C_1,T C_2,T) \tkzDrawLine(C_1,C_2) \tkzMarkRightAngle[fill=teal,opacity=.2,size=.5](C_1,T,C_2) \tkzDrawPoints(C_1,C_2,T) \tkzLabelPoints(C_1,C_2) \tkzLabelPoints[above](T) \tkzLabelSegment[left](C_1,T){r} \tkzLabelSegments[right](C_2,T){$\gamma$} \tkzLabelSegment[below](C_1,C_2){d} \tkzLabelCircle[left=10pt](C_1,T)(180){Circle 1} \tkzLabelCircle[right=10pt](C_2,T)(180){Circle 2} \end{tikzpicture} \hspace*{\fill} % subsection orthogonal_circles_v1 (end) \subsection{Orthogonal circles v2} % (fold) \label{sub:orthogonal_circles_v2} \begin{Verbatim} \begin{tkzelements} scale = .75 z.O = point: new (2,2) z.Op = point: new (-4,1) z.P = point: polar (4,0) C.OP = circle: new (z.O,z.P) C.Oz1 = C.OP : orthogonal_from (z.Op) z.z1 = C.Oz1.through L.OP = line : new (z.O,z.P) C.Opz1 = circle: new (z.Op,z.z1) L.T,L.Tp = C.Opz1 : tangent_from (z.O) z.T = L.T.pb z.Tp = L.Tp.pb L.OOp = line : new (z.O,z.Op) z.M = L.OOp.mid \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle[red](O,P) \tkzDrawCircle[purple](O',z1) \tkzDrawCircle[cyan](M,T) \tkzDrawSegments(O',T O,T' O',T') \tkzDrawSegment[purple](O',T) \tkzDrawSegments[red](O,T O,O') \tkzDrawPoints(O,O',T,T',M) \tkzMarkRightAngle[fill=gray!10](O',T,O) \tkzLabelPoint[below](O){$O$} \tkzLabelPoint[above](T){$T$} \tkzLabelPoint[above](M){$M$} \tkzLabelPoint[below](T'){$T'$} \tkzLabelPoint[above left](O'){$O'$} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = .75 z.O = point: new (2,2) z.Op = point: new (-4,1) z.P = point: polar (4,0) C.OP = circle: new (z.O,z.P) C.Oz1 = C.OP : orthogonal_from (z.Op) z.z1 = C.Oz1.through L.OP = line : new (z.O,z.P) C.Opz1 = circle: new (z.Op,z.z1) L.T,L.Tp = C.Opz1 : tangent_from (z.O) z.T = L.T.pb z.Tp = L.Tp.pb L.OOp = line : new (z.O,z.Op) z.M = L.OOp.mid \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle[red](O,P) \tkzDrawCircle[purple](O',z1) \tkzDrawCircle[cyan](M,T) \tkzDrawSegments(O',T O,T' O',T') \tkzDrawSegment[purple](O',T) \tkzDrawSegments[red](O,T O,O') \tkzDrawPoints(O,O',T,T',M) \tkzMarkRightAngle[fill=gray!10](O',T,O) \tkzLabelPoint[below](O){$O$} \tkzLabelPoint[above](T){$T$} \tkzLabelPoint[above](M){$M$} \tkzLabelPoint[below](T'){$T'$} \tkzLabelPoint[above left](O'){$O'$} \end{tikzpicture} \hspace*{\fill} % subsection orthogonal_circles_v2 (end) % \subsection{Orthogonal circle to two circles} % (fold) \label{sub:orthogonal_circle_to_two_circles} \begin{Verbatim} \begin{tkzelements} z.O = point : new (-1,0) z.B = point : new (0,2) z.Op = point : new (4,-1) z.D = point : new (4,0) C.OB = circle : new (z.O,z.B) C.OpD = circle : new (z.Op,z.D) z.E,z.F = get_points (C.OB : radical_axis (C.OpD)) L.EF = line : new (z.E,z.F) z.M = L.EF : point (.25) L.T,L.Tp = C.OB : tangent_from (z.M) L.K,L.Kp = C.OpD : tangent_from (z.M) z.T = L.T.pb z.K = L.K.pb z.Tp = L.Tp.pb z.Kp = L.Kp.pb \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',D) \tkzDrawLine[cyan](E,F) \tkzDrawLines[add=.5 and .5,orange](O,O' O,T O,T') \tkzDrawSegments[cyan](M,T M,T' M,K M,K') \tkzDrawCircle(M,T) \tkzDrawPoints(O,O',T,M,T',K,K') \tkzLabelPoints(O,O',T,T',M,K,K') \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = 1.25 z.O = point : new (-1,0) z.B = point : new (0,2) z.Op = point : new (4,-1) z.D = point : new (4,0) C.OB = circle : new (z.O,z.B) C.OpD = circle : new (z.Op,z.D) z.E,z.F = get_points (C.OB : radical_axis (C.OpD)) L.EF = line : new (z.E,z.F) z.M = L.EF : point (.25) L.T,L.Tp = C.OB : tangent_from (z.M) L.K,L.Kp = C.OpD : tangent_from (z.M) z.T = L.T.pb z.K = L.K.pb z.Tp = L.Tp.pb z.Kp = L.Kp.pb \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',D) \tkzDrawLine[cyan](E,F) \tkzDrawLines[add=.5 and .5,orange](O,O' O,T O,T') \tkzDrawSegments[cyan](M,T M,T' M,K M,K') \tkzDrawCircle(M,T) \tkzDrawPoints(O,O',T,M,T',K,K') \tkzLabelPoints(O,O',T,T',M,K,K') \end{tikzpicture} \hspace*{\fill} % \subsection{Orthogonal to two circles} % (fold) \subsection{Midcircles} % (fold) \label{sub:midcircles} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (10 , 0) L.AB = line : new (z.A,z.B) z.C = L.AB: gold_ratio () L.AC = line : new (z.A,z.C) L.CB = line : new (z.C,z.B) z.O_0 = L.AB.mid z.O_1 = L.AC.mid z.O_2 = L.CB.mid C.O0B = circle : new (z.O_0,z.B) C.O1C = circle : new (z.O_1,z.C) C.O2C = circle : new (z.O_2,z.B) z.Q = C.O1C : midarc (z.C,z.A) z.P = C.O2C : midarc (z.B,z.C) L.O1O2 = line : new (z.O_1,z.O_2) L.O0O1 = line : new (z.O_0,z.O_1) L.O0O2 = line : new (z.O_0,z.O_2) z.M_0 = L.O1O2 : harmonic_ext (z.C) z.M_1 = L.O0O1 : harmonic_int (z.A) z.M_2 = L.O0O2 : harmonic_int (z.B) L.BQ = line : new (z.B,z.Q) L.AP = line : new (z.A,z.P) z.S = intersection (L.BQ,L.AP) L.CS = line : new (z.C,z.S) C.M1A = circle : new (z.M_1,z.A) C.M2B = circle : new (z.M_2,z.B) z.P_0 = intersection (L.CS,C.O0B) z.P_1 = intersection (C.M2B,C.O1C) z.P_2 = intersection (C.M1A,C.O2C) T.P012 = triangle : new (z.P_0,z.P_1,z.P_2) z.O_4 = T.P012.circumcenter T.CP12 = triangle : new (z.C,z.P_1,z.P_2) z.O_5 = T.CP12.circumcenter z.BN = z.B : north () L.BBN = line : new (z.B,z.BN) L.M1P2 = line : new (z.M_1,z.P_2) z.J = intersection (L.BBN,L.M1P2) L.AP0 = line : new (z.A,z.P_0) L.BP0 = line : new (z.B,z.P_0) C.O4P0 = circle : new (z.O_4,z.P_0) _,z.G = intersection (L.AP0,C.O4P0) z.H = intersection (L.BP0,C.O4P0) z.Ap = z.M_1: symmetry (z.A) z.H_4,z.F,z.E,z.H_0 = L.AB : projection (z.O_4,z.G,z.H,z.P_0) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle[thin,fill=green!10](O_4,P_0) \tkzDrawCircle[purple,fill=purple!10,opacity=.5](O_5,C) \tkzDrawSemiCircles[teal](O_0,B) \tkzDrawSemiCircles[thin,teal,fill=teal!20,opacity=.5](O_1,C O_2,B) \tkzDrawSemiCircles[color = orange](M_2,B) \tkzDrawSemiCircles[color = orange](M_1,A') \tkzDrawArc[purple,delta=0](M_0,P_0)(C) \tkzDrawSegments[very thin](A,B A,P B,Q) \tkzDrawSegments[color=cyan](O_0,P_0 B,J G,J G,O_0 H,O_2) \tkzDrawSegments[ultra thin,purple](M_1,P_0 M_2,P_0 M_1,M_0 M_0,P_1 M_0,P_0 M_1,J) \tkzDrawPoints(A,B,C,P_0,P_2,P_1,M_0,M_1,M_2,J,P,Q,S) \tkzDrawPoints(O_0,O_1,O_2,O_4,O_5,G,H) \tkzMarkRightAngle[size=.2,fill=gray!20,opacity=.4](O_0,P_0,M_0) \tkzLabelPoints[below](A,B,C,M_0,M_1,M_2,O_1,O_2,O_0) \tkzLabelPoints[above](P_0,O_5,O_4) \tkzLabelPoints[above](P_1,J) \tkzLabelPoints[above](P_2,P,Q,S) \tkzLabelPoints[above right](H,E) \tkzLabelPoints[above left](F,G) \tkzLabelPoints[below right](H_0) \tkzLabelCircle[below=4pt,font=\scriptsize](O_1,C)(80){$(\beta)$} \tkzLabelCircle[below=4pt,font=\scriptsize](O_2,B)(80){$(\gamma)$} \tkzLabelCircle[below=4pt,font=\scriptsize](O_0,B)(110){$(\alpha)$} \tkzLabelCircle[left,font=\scriptsize](O_4,P_2)(60){$(\delta)$} \tkzLabelCircle[above left,font=\scriptsize](O_5,C)(40){$(\epsilon)$} \end{tikzpicture} \hspace*{\fill} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (10 , 0) L.AB = line : new (z.A,z.B) z.C = L.AB: gold_ratio () L.AC = line : new (z.A,z.C) L.CB = line : new (z.C,z.B) z.O_0 = L.AB.mid z.O_1 = L.AC.mid z.O_2 = L.CB.mid C.O0B = circle : new (z.O_0,z.B) C.O1C = circle : new (z.O_1,z.C) C.O2C = circle : new (z.O_2,z.B) z.Q = C.O1C : midarc (z.C,z.A) z.P = C.O2C : midarc (z.B,z.C) L.O1O2 = line : new (z.O_1,z.O_2) L.O0O1 = line : new (z.O_0,z.O_1) L.O0O2 = line : new (z.O_0,z.O_2) z.M_0 = L.O1O2 : harmonic_ext (z.C) z.M_1 = L.O0O1 : harmonic_int (z.A) z.M_2 = L.O0O2 : harmonic_int (z.B) L.BQ = line : new (z.B,z.Q) L.AP = line : new (z.A,z.P) z.S = intersection (L.BQ,L.AP) L.CS = line : new (z.C,z.S) C.M1A = circle : new (z.M_1,z.A) C.M2B = circle : new (z.M_2,z.B) z.P_0 = intersection (L.CS,C.O0B) z.P_1 = intersection (C.M2B,C.O1C) z.P_2 = intersection (C.M1A,C.O2C) T.P012 = triangle : new (z.P_0,z.P_1,z.P_2) z.O_4 = T.P012.circumcenter T.CP12 = triangle : new (z.C,z.P_1,z.P_2) z.O_5 = T.CP12.circumcenter z.BN = z.B : north () L.BBN = line : new (z.B,z.BN) L.M1P2 = line : new (z.M_1,z.P_2) z.J = intersection (L.BBN,L.M1P2) L.AP0 = line : new (z.A,z.P_0) L.BP0 = line : new (z.B,z.P_0) C.O4P0 = circle : new (z.O_4,z.P_0) _,z.G = intersection (L.AP0,C.O4P0) z.H = intersection (L.BP0,C.O4P0) z.Ap = z.M_1: symmetry (z.A) z.H_4,z.F,z.E,z.H_0 = L.AB : projection (z.O_4,z.G,z.H,z.P_0) \end{tkzelements} \end{Verbatim} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle[thin,fill=green!10](O_4,P_0) \tkzDrawCircle[purple,fill=purple!10,opacity=.5](O_5,C) \tkzDrawSemiCircles[teal](O_0,B) \tkzDrawSemiCircles[thin,teal,fill=teal!20,opacity=.5](O_1,C O_2,B) \tkzDrawSemiCircles[color = orange](M_2,B) \tkzDrawSemiCircles[color = orange](M_1,A') \tkzDrawArc[purple,delta=0](M_0,P_0)(C) \tkzDrawSegments[very thin](A,B A,P B,Q) \tkzDrawSegments[color=cyan](O_0,P_0 B,J G,J G,O_0 H,O_2) \tkzDrawSegments[ultra thin,purple](M_1,P_0 M_2,P_0 M_1,M_0 M_0,P_1 M_0,P_0 M_1,J) \tkzDrawPoints(A,B,C,P_0,P_2,P_1,M_0,M_1,M_2,J,P,Q,S) \tkzDrawPoints(O_0,O_1,O_2,O_4,O_5,G,H) \tkzMarkRightAngle[size=.2,fill=gray!20,opacity=.4](O_0,P_0,M_0) \tkzLabelPoints[below](A,B,C,M_0,M_1,M_2,O_1,O_2,O_0) \tkzLabelPoints[above](P_0,O_5,O_4) \tkzLabelPoints[above](P_1,J) \tkzLabelPoints[above](P_2,P,Q,S) \tkzLabelPoints[above right](H,E) \tkzLabelPoints[above left](F,G) \tkzLabelPoints[below right](H_0) \tkzLabelCircle[below=4pt,font=\scriptsize](O_1,C)(80){$(\beta)$} \tkzLabelCircle[below=4pt,font=\scriptsize](O_2,B)(80){$(\gamma)$} \tkzLabelCircle[below=4pt,font=\scriptsize](O_0,B)(110){$(\alpha)$} \tkzLabelCircle[left,font=\scriptsize](O_4,P_2)(60){$(\delta)$} \tkzLabelCircle[above left,font=\scriptsize](O_5,C)(40){$(\epsilon)$} \end{tikzpicture} \end{Verbatim} % subsection midcircles (end) \subsection{Pencil v1} % (fold) \label{sub:pencil_v1} \begin{Verbatim} \begin{tkzelements} scale = .75 z.A = point : new (0,2) z.B = point : new (0,-2) z.C_0 = point : new (-3,0) z.C_1 = point : new (2,0) z.C_3 = point : new (2.5,0) z.C_5 = point : new (1,0) L.BA = line : new (z.B,z.A) z.M_0 = L.BA : point (1.25) z.M_1 = L.BA : point (1.5) C.C0A = circle : new (z.C_0,z.A) z.x,z.y = get_points (C.C0A : orthogonal_from (z.M_0)) z.xp,z.yp = get_points (C.C0A : orthogonal_from (z.M_1)) z.O = L.BA.mid \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_0,A C_1,A C_3,A C_5,A) \tkzDrawCircles[thick,color=red](M_0,x M_1,x') \tkzDrawCircles[thick,color=blue](O,A) \tkzDrawLines(C_0,C_1 B,M_1) \tkzDrawPoints(A,B,C_0,C_1,M_0,M_1,x,y) \tkzLabelPoints[below right](A,B,C_0,C_1,M_0,M_1,x,y) \tkzLabelLine[pos=1.25,right]( M_0,M_1){$(\Delta)$} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale=.75 z.A = point : new (0,2) z.B = point : new (0,-2) z.C_0 = point : new (-3,0) z.C_1 = point : new (2,0) z.C_3 = point : new (2.5,0) z.C_5 = point : new (1,0) L.BA = line : new (z.B,z.A) z.M_0 = L.BA : point (1.25) z.M_1 = L.BA : point (1.5) C.C0A = circle : new (z.C_0,z.A) z.x,z.y = get_points (C.C0A : orthogonal_from (z.M_0)) z.xp,z.yp = get_points (C.C0A : orthogonal_from (z.M_1)) z.O = L.BA.mid \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_0,A C_1,A C_3,A C_5,A) \tkzDrawCircles[thick,color=red](M_0,x M_1,x') \tkzDrawCircles[thick,color=blue](O,A) \tkzDrawLines(C_0,C_1 B,M_1) \tkzDrawPoints(A,B,C_0,C_1,M_0,M_1,x,y) \tkzLabelPoints[below right](A,B,C_0,C_1,M_0,M_1,x,y) \tkzLabelLine[pos=1.25,right]( M_0,M_1){$(\Delta)$} \end{tikzpicture} \hspace*{\fill} % subsection pencil_v1 (end) \subsection{Pencil v2} % (fold) \label{sub:pencil_v2} \begin{Verbatim} \begin{tkzelements} scale=.75 z.A = point : new (0,0) z.B = point : new (1,0) z.C_0 = point : new (-2,0) z.C_1 = point : new (4,0) C.C0A = circle : new (z.C_0,z.A) C.C1B = circle : new (z.C_1,z.B) L.EF = C.C0A : radical_axis (C.C1B) z.M_0 = L.EF : point (.4) z.M_1 = L.EF : point (.1) z.M_2 = L.EF : point (.6) C.orth0 = C.C0A : orthogonal_from (z.M_0) C.orth1 = C.C0A : orthogonal_from (z.M_1) C.orth2 = C.C0A : orthogonal_from (z.M_2) z.u = C.orth0.through z.v = C.orth1.through z.t = C.orth2.through \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_0,A C_1,B) \tkzDrawCircles[thick,color=red](M_0,u M_1,v M_2,t) \tkzDrawLines[add= .75 and .75](C_0,C_1 M_0,M_1) \tkzDrawPoints(A,B,C_0,C_1,M_0,M_1,M_2) \tkzLabelPoints[below right](A,B,C_0,C_1,M_0,M_1,M_2) \tkzLabelLine[pos=2,right]( M_0,M_1){$(\Delta)$} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale=.75 z.A = point : new (0,0) z.B = point : new (1,0) z.C_0 = point : new (-2,0) z.C_1 = point : new (4,0) C.C0A = circle : new (z.C_0,z.A) C.C1B = circle : new (z.C_1,z.B) L.EF = C.C0A : radical_axis (C.C1B) z.M_0 = L.EF : point (.4) z.M_1 = L.EF : point (.1) z.M_2 = L.EF : point (.6) C.orth0 = C.C0A : orthogonal_from (z.M_0) C.orth1 = C.C0A : orthogonal_from (z.M_1) C.orth2 = C.C0A : orthogonal_from (z.M_2) z.u = C.orth0.through z.v = C.orth1.through z.t = C.orth2.through \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_0,A C_1,B) \tkzDrawCircles[thick,color=red](M_0,u M_1,v M_2,t) \tkzDrawLines[add= .75 and .75](C_0,C_1 M_0,M_1) \tkzDrawPoints(A,B,C_0,C_1,M_0,M_1,M_2) \tkzLabelPoints[below right](A,B,C_0,C_1,M_0,M_1,M_2) \tkzLabelLine[pos=2,right]( M_0,M_1){$(\Delta)$} \end{tikzpicture} \hspace*{\fill} %subsection pencil_v2 (end) \subsection{Power v1} % (fold) \label{sub:power_v1} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.A = point : new (2,-2) z.M = point : new (-6,0) L.AM = line : new (z.A,z.M) C.OA = circle : new (z.O,z.A) z.Ap = C.OA : antipode (z.A) z.B = intersection (L.AM, C.OA) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzMarkRightAngle[fill=gray!10](A',B,M) \tkzDrawSegments(M,O A,A' A',B) \tkzDrawPoints(O,A,A',M,B) \tkzLabelPoints(O,A,A',M,B) \tkzDrawSegments[-Triangle](M,A M,A') \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = 1 z.O = point : new (0,0) z.A = point : new (2,-2) z.M = point : new (-6,0) L.AM = line : new (z.A,z.M) C.OA = circle : new (z.O,z.A) z.Ap = C.OA : antipode (z.A) z.B = intersection (L.AM, C.OA) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzMarkRightAngle[fill=gray!10](A',B,M) \tkzDrawSegments(M,O A,A' A',B) \tkzDrawPoints(O,A,A',M,B) \tkzLabelPoints(O,A,A',M,B) \tkzDrawSegments[-Triangle](M,A M,A') \end{tikzpicture} \end{minipage} % subsection power_v1 (end) \subsection{Power v2} % (fold) \label{sub:power_v2} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.O = point : new (0,0) z.A = point : new (2,2) z.M = point : new (-1.5,0) L.AM = line : new (z.A,z.M) C.OA = circle : new (z.O,z.A) z.Ap = C.OA : antipode (z.A) _,z.B = intersection (L.AM, C.OA) z.m = z.M : north(1) L.mM = line : new (z.m,z.M) z.U,z.V = intersection (L.mM,C.OA) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzMarkRightAngle[fill=gray!10](A',B,M) \tkzDrawSegments(M,O A,A' A',B A,B U,V) \tkzDrawPoints(O,A,A',M,B,U,V,m) \tkzLabelPoints(O,A,M,U,V,m) \tkzLabelPoints[below left](A',B) \tkzDrawSegments(M,A M,A') \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = 1 z.O = point : new (0,0) z.A = point : new (2,2) z.M = point : new (-1.5,0) L.AM = line : new (z.A,z.M) C.OA = circle : new (z.O,z.A) z.Ap = C.OA : antipode (z.A) _,z.B = intersection (L.AM, C.OA) z.m = z.M : north(1) L.mM = line : new (z.m,z.M) z.U,z.V = intersection (L.mM,C.OA) \end{tkzelements} \hspace*{\fill}\begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzMarkRightAngle[fill=gray!10](A',B,M) \tkzDrawSegments(M,O A,A' A',B A,B U,V) \tkzDrawPoints(O,A,A',M,B,U,V,m) \tkzLabelPoints(O,A,M,U,V,m) \tkzLabelPoints[below left](A',B) \tkzDrawSegments(M,A M,A') \end{tikzpicture} \end{minipage} % subsection power_v2 (end) \subsection{Reim v1} % (fold) \label{sub:reim_v1} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0,0) z.E = point: new (-2,2) C.AE = circle : new (z.A,z.E) z.C = C.AE : point (0.65) z.D = C.AE : point (0.5) z.F = C.AE : point (0.30) L.EC = line: new (z.E,z.C) z.H = L.EC : point (1.5) T.CDH = triangle : new (z.C,z.D,z.H) z.B = T.CDH.circumcenter C.BD = circle : new (z.B,z.D) L.FD = line: new (z.F,z.D) z.G = intersection (L.FD,C.BD) z.O = intersection (L.EC,L.FD) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,E B,H) \tkzDrawSegments(E,D C,F) \tkzDrawLines(E,O F,O) \tkzDrawLines[red](E,F H,G) \tkzDrawPoints(A,...,H,O) \tkzLabelPoints(A,B,D,F,G,O) \tkzLabelPoints[above](E,C,H) \tkzMarkAngles[size=.5](E,C,F E,D,F) \tkzFillAngles[green!40,opacity=.4,size=.5](E,C,F E,D,F) \tkzMarkAngles[size=.5](F,C,H G,D,E) \tkzFillAngles[red!40,opacity=.4,size=.5](F,C,H G,D,E) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.A = point: new (0,0) z.E = point: new (-2,2) C.AE = circle : new (z.A,z.E) z.C = C.AE : point (0.65) z.D = C.AE : point (0.5) z.F = C.AE : point (0.30) L.EC = line: new (z.E,z.C) z.H = L.EC : point (1.5) T.CDH = triangle : new (z.C,z.D,z.H) z.B = T.CDH.circumcenter C.BD = circle : new (z.B,z.D) L.FD = line: new (z.F,z.D) z.G = intersection (L.FD,C.BD) z.O = intersection (L.EC,L.FD) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,E B,H) \tkzDrawSegments(E,D C,F) \tkzDrawLines(E,O F,O) \tkzDrawLines[red](E,F H,G) \tkzDrawPoints(A,...,H,O) \tkzLabelPoints(A,B,D,F,G,O) \tkzLabelPoints[above](E,C,H) \tkzMarkAngles[size=.5](E,C,F E,D,F) \tkzFillAngles[green!40,opacity=.4,size=.5](E,C,F E,D,F) \tkzMarkAngles[size=.5](F,C,H G,D,E) \tkzFillAngles[red!40,opacity=.4,size=.5](F,C,H G,D,E) \end{tikzpicture} \hspace*{\fill} % subsection reim_v1 (end) \subsection{Reim v2} % (fold) \label{sub:reim_v2} \begin{Verbatim} \begin{tkzelements} scale = .6 z.A = point: new (0,0) z.B = point: new (10,0) z.C = point: new (4,0) C.AC = circle: new (z.A,z.C) z.c,z.cp = get_points (C.AC: tangent_at (z.C)) z.M = C.AC: point (0.6) L.MC = line: new (z.M,z.C) C.BC = circle: new (z.B,z.C) z.N = intersection (L.MC,C.BC) z.m,z.mp = get_points (C.AC: tangent_at (z.M)) z.n,z.np = get_points (C.BC: tangent_at (z.N)) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,C B,C) \tkzDrawLines[new,add=1 and 1](M,m N,n C,c) \tkzDrawSegment(M,N) \tkzDrawPoints(A,B,C,M,N) \tkzLabelPoints[below right](A,B,C,M,N) \tkzFillAngles[blue!30,opacity=.3](m',M,C N,C,c' M,C,c n',N,C) \tkzLabelCircle[below=4pt,font=\scriptsize](A,C)(90){$(\alpha)$} \tkzLabelCircle[left=4pt,font=\scriptsize](B,C)(-90){$(\beta)$} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = .4 z.A = point: new (0,0) z.B = point: new (10,0) z.C = point: new (4,0) C.AC = circle: new (z.A,z.C) z.c,z.cp = get_points (C.AC: tangent_at (z.C)) z.M = C.AC: point (0.6) L.MC = line: new (z.M,z.C) C.BC = circle: new (z.B,z.C) z.N = intersection (L.MC,C.BC) z.m,z.mp = get_points (C.AC: tangent_at (z.M)) z.n,z.np = get_points (C.BC: tangent_at (z.N)) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,C B,C) \tkzDrawLines[new,add=1 and 1](M,m N,n C,c) \tkzDrawSegment(M,N) \tkzDrawPoints(A,B,C,M,N) \tkzLabelPoints[below right](A,B,C,M,N) \tkzFillAngles[blue!30,opacity=.3](m',M,C N,C,c' M,C,c n',N,C) \tkzLabelCircle[below=4pt,font=\scriptsize](A,C)(90){$(\alpha)$} \tkzLabelCircle[left=4pt,font=\scriptsize](B,C)(-90){$(\beta)$} \end{tikzpicture} \hspace*{\fill} % subsection reim_v2 (end) \subsection{Reim v3} % (fold) \label{sub:reim_v3} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0,0) z.B = point: new (8,0) z.C = point: new (2,6) L.AB = line : new (z.A,z.B) L.AC = line : new (z.A,z.C) L.BC = line : new (z.B,z.C) z.I = L.BC : point (0.75) z.J = L.AC : point (0.4) z.K = L.AB : point (0.5) T.AKJ = triangle : new (z.A,z.K,z.J) T.BIK = triangle : new (z.B,z.I,z.K) T.CIJ = triangle : new (z.C,z.I,z.J) z.x = T.AKJ.circumcenter z.y = T.BIK.circumcenter z.z = T.CIJ.circumcenter C.xK = circle: new (z.x,z.K) C.yK = circle: new (z.y,z.K) z.O,_ = intersection (C.xK,C.yK) C.zO = circle: new (z.z,z.O) L.KO = line: new (z.K,z.O) z.D = intersection (L.KO,C.zO) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawSegments(K,D D,C) \tkzDrawPolygon[teal](A,B,C) \tkzDrawCircles[orange](x,A y,B z,C) \tkzDrawPoints[fill=white](A,B,C,I,J,K,D) \tkzLabelPoints[below](A,B,J,K,O) \tkzLabelPoints[above](C,D,I) \tkzDrawPoints[fill=black](O) \tkzLabelCircle[below=4pt,font=\scriptsize](x,A)(20){$(\alpha)$} \tkzLabelCircle[left=4pt,font=\scriptsize](y,B)(60){$(\beta)$} \tkzLabelCircle[below=4pt,font=\scriptsize](z,C)(60){$(\gamma)$} \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = .75 z.A = point: new (0,0) z.B = point: new (8,0) z.C = point: new (2,6) L.AB = line : new (z.A,z.B) L.AC = line : new (z.A,z.C) L.BC = line : new (z.B,z.C) z.I = L.BC : point (0.75) z.J = L.AC : point (0.4) z.K = L.AB : point (0.5) T.AKJ = triangle : new (z.A,z.K,z.J) T.BIK = triangle : new (z.B,z.I,z.K) T.CIJ = triangle : new (z.C,z.I,z.J) z.x = T.AKJ.circumcenter z.y = T.BIK.circumcenter z.z = T.CIJ.circumcenter C.xK = circle: new (z.x,z.K) C.yK = circle: new (z.y,z.K) z.O,_ = intersection (C.xK,C.yK) C.zO = circle: new (z.z,z.O) L.KO = line: new (z.K,z.O) z.D = intersection (L.KO,C.zO) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawSegments(K,D D,C) \tkzDrawPolygon[teal](A,B,C) \tkzDrawCircles[orange](x,A y,B z,C) \tkzDrawPoints[fill=white](A,B,C,I,J,K,D) \tkzLabelPoints[below](A,B,J,K,O) \tkzLabelPoints[above](C,D,I) \tkzDrawPoints[fill=black](O) \tkzLabelCircle[below=4pt,font=\scriptsize](x,A)(20){$(\alpha)$} \tkzLabelCircle[left=4pt,font=\scriptsize](y,B)(60){$(\beta)$} \tkzLabelCircle[below=4pt,font=\scriptsize](z,C)(60){$(\gamma)$} \end{tikzpicture} \hspace*{\fill} % subsection reim_v3 (end) \subsection{Tangent and circle} % (fold) \label{sub:tangent_and_circle} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point: new (1,0) z.B = point: new (2,2) z.E = point: new (5,-4) L.AE = line : new (z.A,z.E) C.AB = circle: new (z.A , z.B) z.S = C.AB.south z.M = L.AE.mid L.Ti,L.Tj = C.AB: tangent_from (z.E) z.i = L.Ti.pb z.j = L.Tj.pb z.k,z.l = get_points (C.AB: tangent_at (z.B)) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,B M,A) \tkzDrawPoints(A,B,E,i,j,M,S) \tkzDrawLines(E,i E,j k,l) \tkzLabelPoints[right,font=\small](A,B,E,S,M) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} z.A = point: new (1,0) z.B = point: new (2,2) z.E = point: new (5,-4) L.AE = line : new (z.A,z.E) C.AB = circle: new (z.A , z.B) z.S = C.AB.south z.M = L.AE.mid L.Ti,L.Tj = C.AB: tangent_from (z.E) z.i = L.Ti.pb z.j = L.Tj.pb z.k,z.l = get_points (C.AB: tangent_at (z.B)) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,B M,A) \tkzDrawPoints(A,B,E,i,j,M,S) \tkzDrawLines(E,i E,j k,l) \tkzLabelPoints[right,font=\small](A,B,E,S,M) \end{tikzpicture} \end{minipage} % subsection tangent_and_circle (end) \subsection{Homothety} % (fold) \label{sub:homothety} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0,0) z.B = point: new (1,2) z.E = point: new (-3,2) z.C,z.D = z.E : homothety(2,z.A,z.B) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C,E,D) \tkzLabelPoints(A,B,C,E) \tkzDrawCircles(A,B C,D) \tkzDrawLines(E,C E,D) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} scale = .6 z.A = point: new (0,0) z.B = point: new (1,2) z.E = point: new (-3,2) z.C,z.D = z.E : homothety(2,z.A,z.B) \end{tkzelements} \hspace*{\fill}\begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C,E,D) \tkzLabelPoints(A,B,C,E) \tkzDrawCircles(A,B C,D) \tkzDrawLines(E,C E,D) \end{tikzpicture} \end{minipage} % subsection homothety (end) \subsection{Tangent and chord} % (fold) \label{sub:tangent_and_chord} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} scale = .8 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (1 , 5) z.Bp = point: new (2 , 0) T.ABC = triangle: new (z.A,z.B,z.C) L.AB = line: new (z.A,z.B) z.O = T.ABC.circumcenter C.OA = circle: new (z.O,z.A) z.D = C.OA: point (4.5) L.AO = line: new (z.A,z.O) z.b1,z.b2 = get_points (C.OA: tangent_at (z.B)) z.H = L.AB: projection (z.O) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzDrawPolygon(A,B,C) \tkzDrawSegments[new](A,O B,O O,H A,D D,B) \tkzDrawLine(b1,b2) \tkzDrawPoints(A,B,C,D,H,O) \tkzFillAngles[green!20,opacity=.3](H,O,B A,C,B A,B,b1) \tkzFillAngles[teal!20,opacity=.3](A,D,B b2,B,A) \tkzLabelPoints(A,B,C,D,H,O) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} scale = 0.75 z.A = point: new (0 , 0) z.B = point: new (6 , 0) z.C = point: new (1 , 5) z.Bp = point: new (2 , 0) T.ABC = triangle: new (z.A,z.B,z.C) L.AB = line: new (z.A,z.B) z.O = T.ABC.circumcenter C.OA = circle: new (z.O,z.A) z.D = C.OA: point (4.5) L.AO = line: new (z.A,z.O) z.b1,z.b2 = get_points (C.OA: tangent_at (z.B)) z.H = L.AB: projection (z.O) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzDrawPolygon(A,B,C) \tkzDrawSegments[new](A,O B,O O,H A,D D,B) \tkzDrawLine(b1,b2) \tkzDrawPoints(A,B,C,D,H,O) \tkzFillAngles[green!20,opacity=.3](H,O,B A,C,B A,B,b1) \tkzFillAngles[teal!20,opacity=.3](A,D,B b2,B,A) \tkzLabelPoints(A,B,C,D,H,O) \end{tikzpicture} \end{minipage} % subsection tangent_and_chord (end) \subsection{Three chords} % (fold) \label{sub:three_chords} \begin{Verbatim} \begin{tkzelements} z.O = point: new (0 , 0) z.B = point: new (0 , 2) z.P = point: new (1 , -.5) C.OB = circle : new (z.O,z.B) C.PB = circle : new (z.P,z.B) _,z.A = intersection (C.OB,C.PB) z.D = C.PB: point(0.85) z.C = C.PB: point(0.5) z.E = C.OB: point(0.6) L.AB = line : new (z.A,z.B) L.CD = line : new (z.C,z.D) z.G = intersection (L.AB,L.CD) L.GE = line : new (z.G,z.E) z.F,_ = intersection (L.GE,C.OB) T.CDE = triangle: new (z.C,z.D,z.E) T.BFD = triangle: new (z.B,z.F,z.D) z.w = T.CDE.circumcenter z.x = T.BFD.circumcenter L.GB = line : new (z.G,z.B) L.GE = line : new (z.G,z.E) L.GD = line : new (z.G,z.D) C.xB = circle : new (z.x,z.B) C.xF = circle : new (z.x,z.F) C.xD = circle : new (z.x,z.D) z.Ap = intersection (L.GB,C.xB) z.Ep,_ = intersection (L.GE,C.xF) z.Cp,_ = intersection (L.GD,C.xD) \end{tkzelements} \end{Verbatim} \begin{tkzelements} z.O = point: new (0 , 0) z.B = point: new (0 , 2) z.P = point: new (1 , -.5) C.OB = circle : new (z.O,z.B) C.PB = circle : new (z.P,z.B) _,z.A = intersection (C.OB,C.PB) z.D = C.PB: point(0.85) z.C = C.PB: point(0.5) z.E = C.OB: point(0.6) L.AB = line : new (z.A,z.B) L.CD = line : new (z.C,z.D) z.G = intersection (L.AB,L.CD) L.GE = line : new (z.G,z.E) z.F,_ = intersection (L.GE,C.OB) T.CDE = triangle: new (z.C,z.D,z.E) T.BFD = triangle: new (z.B,z.F,z.D) z.w = T.CDE.circumcenter z.x = T.BFD.circumcenter L.GB = line : new (z.G,z.B) L.GE = line : new (z.G,z.E) L.GD = line : new (z.G,z.D) C.xB = circle : new (z.x,z.B) C.xF = circle : new (z.x,z.F) C.xD = circle : new (z.x,z.D) z.Ap = intersection (L.GB,C.xB) z.Ep,_ = intersection (L.GE,C.xF) z.Cp,_ = intersection (L.GD,C.xD) \end{tkzelements} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B) \tkzDrawCircles[cyan](P,B) \tkzDrawCircles[red](w,E) \tkzDrawCircles[new](x,F) \tkzDrawSegments(A,G E,G C,G) \tkzDrawPolygons[new](A,E,C A',E',C') \tkzDrawPoints(A,...,G,A',E',C',O,P) \begin{scope}[font=\scriptsize] \tkzLabelPoints(A,...,F) \tkzLabelPoints[above left](G,A',E',C') \tkzLabelCircle[left](O,B)(30){$(\beta)$} \tkzLabelCircle[below](P,A)(40){$(\gamma)$} \tkzLabelCircle[right](w,C)(90){$(\alpha)$} \tkzLabelCircle[left](x,B)(-230){$((\delta))$} \end{scope} \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \hspace*{\fill} \begin{tikzpicture}[scale=.75] \tkzGetNodes \tkzDrawCircles(O,B) \tkzDrawCircles[cyan](P,B) \tkzDrawCircles[red](w,E) \tkzDrawCircles[new](x,F) \tkzDrawSegments(A,G E,G C,G) \tkzDrawPolygons[new](A,E,C A',E',C') \tkzDrawPoints(A,...,G,A',E',C',O,P) \begin{scope}[font=\scriptsize] \tkzLabelPoints(A,...,F) \tkzLabelPoints[above left](G,A',E',C') \tkzLabelCircle[left](O,B)(30){$(\beta)$} \tkzLabelCircle[below](P,A)(40){$(\gamma)$} \tkzLabelCircle[right](w,C)(90){$(\alpha)$} \tkzLabelCircle[left](x,B)(-230){$((\delta))$} \end{scope} \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection three_chords (end) \subsection{Three tangents} % (fold) \label{sub:three_tangents} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.A = point: new (-1 , 0) z.C = point: new (4 , -1.5) z.E = point: new (1 , -1) z.F = point: new (1.5 , 2.5) T.AEF = triangle : new (z.A,z.E,z.F) T.CEF = triangle : new (z.C,z.E,z.F) z.w = T.AEF.circumcenter z.x = T.CEF.circumcenter C.wE = circle : new (z.w,z.E) C.xE = circle : new (z.x,z.E) L.Aw = line : new (z.A,z.w) L.Cx = line : new (z.C,z.x) z.G = intersection (L.Aw,L.Cx) L.TA = C.wE : tangent_at (z.A) L.TC = C.xE : tangent_at (z.C) z.I = intersection (L.TA,L.TC) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(w,E) \tkzDrawCircles[cyan](x,E) \tkzDrawCircles[red](G,A) \tkzDrawLines(A,I C,I F,I) \tkzDrawPoints(A,C,E,F) \tkzLabelPoints[right](A) \tkzLabelPoints[above right](E,F) \tkzLabelPoints[below](C) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .75 z.A = point: new (-1 , 0) z.C = point: new (4 , -1.5) z.E = point: new (1 , -1) z.F = point: new (1.5 , 2.5) T.AEF = triangle : new (z.A,z.E,z.F) T.CEF = triangle : new (z.C,z.E,z.F) z.w = T.AEF.circumcenter z.x = T.CEF.circumcenter C.wE = circle : new (z.w,z.E) C.xE = circle : new (z.x,z.E) L.Aw = line : new (z.A,z.w) L.Cx = line : new (z.C,z.x) z.G = intersection (L.Aw,L.Cx) L.TA = C.wE : tangent_at (z.A) L.TC = C.xE : tangent_at (z.C) z.I = intersection (L.TA,L.TC) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(w,E) \tkzDrawCircles[cyan](x,E) \tkzDrawCircles[red](G,A) \tkzDrawLines(A,I C,I F,I) \tkzDrawPoints(A,C,E,F) \tkzLabelPoints[right](A) \tkzLabelPoints[above right](E,F) \tkzLabelPoints[below](C) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection three_tangents (end) \subsection{Midarc} % (fold) \label{sub:midarc} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.A = point: new (-1,0) z.B = point: new (2,4) C.AB = circle: new (z.A,z.B) z.C = z.A: rotation (math.pi/3,z.B) z.D = C.AB: midarc (z.B,z.C) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C) \tkzDrawCircles(A,B) \tkzDrawPoints(A,...,D) \tkzLabelPoints(A,...,D) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = .5 z.A = point: new (-1,0) z.B = point: new (2,4) C.AB = circle: new (z.A,z.B) z.C = z.A: rotation (math.pi/3,z.B) z.D = C.AB: midarc (z.B,z.C) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C) \tkzDrawCircles(A,B) \tkzDrawPoints(A,...,D) \tkzLabelPoints(A,...,D) \end{tikzpicture} \hspace*{\fill} \end{minipage} % subsection midarc (end) \subsection{Lemoine Line without macro} % (fold) \label{sub:lemoine_line_without_macro} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} scale = 1.6 z.A = point: new (1,0) z.B = point: new (5,2) z.C = point: new (1.2,2) T = triangle: new(z.A,z.B,z.C) z.O = T.circumcenter L.AB = line: new (z.A,z.B) L.AC = line: new (z.A,z.C) L.BC = line: new (z.B,z.C) C.OA = circle: new (z.O,z.A) z.Ar,z.Al = get_points (C.OA: tangent_at (z.A)) z.Br,z.Bl = get_points (C.OA: tangent_at (z.B)) z.Cr,z.Cl = get_points (C.OA: tangent_at (z.C)) L.tA = line: new (z.Ar,z.Al) L.tB = line: new (z.Br,z.Bl) L.tC = line: new (z.Cr,z.Cl) z.P = intersection (L.tA,L.BC) z.Q = intersection (L.tB,L.AC) z.R = intersection (L.tC,L.AB) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon[teal](A,B,C) \tkzDrawCircle(O,A) \tkzDrawPoints(A,B,C,P,Q,R) \tkzLabelPoints(A,B,C,P,Q,R) \tkzDrawLine[blue](Q,R) \tkzDrawLines[red](Ar,Al Br,Q Cr,Cl) \tkzDrawSegments(A,R C,P C,Q) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} scale = 0.75 z.A = point: new (1,0) z.B = point: new (5,2) z.C = point: new (1.2,2) T = triangle: new(z.A,z.B,z.C) z.O = T.circumcenter L.AB = line: new (z.A,z.B) L.AC = line: new (z.A,z.C) L.BC = line: new (z.B,z.C) C.OA = circle: new (z.O,z.A) z.Ar,z.Al = get_points (C.OA: tangent_at (z.A)) z.Br,z.Bl = get_points (C.OA: tangent_at (z.B)) z.Cr,z.Cl = get_points (C.OA: tangent_at (z.C)) L.tA = line: new (z.Ar,z.Al) L.tB = line: new (z.Br,z.Bl) L.tC = line: new (z.Cr,z.Cl) z.P = intersection (L.tA,L.BC) z.Q = intersection (L.tB,L.AC) z.R = intersection (L.tC,L.AB) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon[teal](A,B,C) \tkzDrawCircle(O,A) \tkzDrawPoints(A,B,C,P,Q,R) \tkzLabelPoints(A,B,C,P,Q,R) \tkzDrawLine[blue](Q,R) \tkzDrawLines[red](Ar,Al Br,Q Cr,Cl) \tkzDrawSegments(A,R C,P C,Q) \end{tikzpicture} \end{minipage} % subsection lemoine_line_without_macro (end) \subsection{First Lemoine circle} % (fold) \label{sub:first_lemoine_circle} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.A = point: new (1,1) z.B = point: new (5,1) z.C = point: new (2,3) T = triangle: new (z.A,z.B,z.C) z.O = T.circumcenter z.o,z.w = get_points (T : first_lemoine_circle ()) z.L = T : lemoine_point () \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(A,B,C) \tkzDrawPoints(A,B,C,o,w,O,L) \tkzLabelPoints(A,B,C,o,w,O,L) \tkzDrawCircles(o,w O,A) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} z.A = point: new (1,1) z.B = point: new (5,1) z.C = point: new (2,3) T = triangle: new (z.A,z.B,z.C) z.O = T.circumcenter z.o,z.w = get_points(T:first_lemoine_circle()) z.L = T : lemoine_point () \end{tkzelements} \hspace*{\fill} \begin{tikzpicture}[scale = 1.25] \tkzGetNodes \tkzDrawPolygons(A,B,C) \tkzDrawPoints(A,B,C,o,w,O,L) \tkzLabelPoints(A,B,C,o,w,O,L) \tkzDrawCircles(o,w O,A) \end{tikzpicture} \end{minipage} % subsection first_lemoine_circle (end) \subsection{First and second Lemoine circles} % (fold) \label{sub:first_and_second_lemoine_circles} \begin{Verbatim} \begin{tkzelements} scale = 2 z.a = point: new (0,0) z.b = point: new (5,0) z.c = point: new (2,3) T = triangle: new (z.a,z.b,z.c) z.O = T.circumcenter z.o,z.p = get_points (T : first_lemoine_circle ()) L.ab = line : new (z.a,z.b) L.ca = line : new (z.c,z.a) L.bc = line : new (z.b,z.c) z.L,z.x = get_points (T : second_lemoine_circle ()) C.first_lemoine = circle : new (z.o,z.p) z.y1,z.y2 = intersection (L.ab,C.first_lemoine) z.y5,z.y6 = intersection (L.bc,C.first_lemoine) z.y3,z.y4 = intersection (L.ca,C.first_lemoine) C.second_lemoine = circle : new (z.L,z.x) z.x1,z.x2 = intersection (L.ab,C.second_lemoine) z.x3,z.x4 = intersection (L.bc,C.second_lemoine) z.x5,z.x6 = intersection (L.ca,C.second_lemoine) L.y1y6 = line : new (z.y1,z.y6) L.y4y5 = line : new (z.y4,z.y5) L.y2y3 = line : new (z.y2,z.y3) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(a,b,c y1,y2,y3,y4,y5,y6) \tkzDrawPoints(x1,x2,x3,x4,x5,x6,L) \tkzDrawPoints(a,b,c,o,O,y1,y2,y3,y4,y5,y6) \tkzLabelPoints[below right](a,b,c,o,O,y1,y2,y3,y4,y5,y6) \tkzLabelPoints[below left](x1,x2,x3,x4,x5,x6) \tkzLabelPoints[above](L) \tkzDrawCircles(L,x o,p O,a) \tkzDrawSegments(L,O x1,x4 x2,x5 x3,x6) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} scale = 2 z.a = point: new (0,0) z.b = point: new (5,0) z.c = point: new (2,3) T = triangle: new (z.a,z.b,z.c) z.O = T.circumcenter z.o,z.p = get_points (T : first_lemoine_circle ()) L.ab = line : new (z.a,z.b) L.ca = line : new (z.c,z.a) L.bc = line : new (z.b,z.c) z.L,z.x = get_points (T : second_lemoine_circle ()) C.first_lemoine = circle : new (z.o,z.p) z.y1,z.y2 = intersection (L.ab,C.first_lemoine) z.y5,z.y6 = intersection (L.bc,C.first_lemoine) z.y3,z.y4 = intersection (L.ca,C.first_lemoine) C.second_lemoine = circle : new (z.L,z.x) z.x1,z.x2 = intersection (L.ab,C.second_lemoine) z.x3,z.x4 = intersection (L.bc,C.second_lemoine) z.x5,z.x6 = intersection (L.ca,C.second_lemoine) L.y1y6 = line : new (z.y1,z.y6) L.y4y5 = line : new (z.y4,z.y5) L.y2y3 = line : new (z.y2,z.y3) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture}[scale = .75] \tkzGetNodes \tkzDrawPolygons(a,b,c y1,y2,y3,y4,y5,y6) \tkzDrawPoints(x1,x2,x3,x4,x5,x6,L) \tkzDrawPoints(a,b,c,o,O,y1,y2,y3,y4,y5,y6) \tkzLabelPoints[below right](a,b,c,o,O,y1,y2,y3,y4,y5,y6) \tkzLabelPoints[below left](x1,x2,x3,x4,x5,x6) \tkzLabelPoints[above](L) \tkzDrawCircles(L,x o,p O,a) \tkzDrawSegments(L,O x1,x4 x2,x5 x3,x6) \end{tikzpicture} \hspace*{\fill} % subsection first_and_second_lemoine_circles (end) \subsection{Inversion} % (fold) \label{sub:inversion} \begin{tkzelements} scale = .75 z.A = point: new (-1,0) z.B = point: new (2,2) z.C = point: new (2,4) z.E = point: new (1,6) C.AC = circle: new (z.A,z.C) L.Tt1,L.Tt2 = C.AC: tangent_from (z.E) z.t1 = L.Tt1.pb z.t2 = L.Tt2.pb L.AE = line: new (z.A,z.E) z.H = L.AE : projection (z.t1) z.Bp, z.Ep, z.Cp = C.AC: inversion ( z.B, z.E, z.C ) \end{tkzelements} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point: new (-1,0) z.B = point: new (2,2) z.C = point: new (2,4) z.E = point: new (1,6) C.AC = circle: new (z.A,z.C) L.Tt1, L.Tt2 = C.AC: tangent_from (z.E) z.t1 = L.Tt1.pb z.t2 = L.Tt2.pb L.AE = line: new (z.A,z.E) z.H = L.AE : projection (z.t1) z.Bp, z.Ep, z.Cp = C.AC: inversion ( z.B, z.E, z.C ) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(A,B,C) \tkzDrawCircles(A,C A,B) \tkzDrawLines(A,B' E,t1 E,t2 t1,t2 A,E) \tkzDrawPoints(A,B,C,E,t1,t2,H,B',E') \tkzLabelPoints(A,B,C,E,t1,t2,B',E') \tkzLabelPoints[above](C') \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tikzpicture}[scale=.75] \tkzGetNodes \tkzDrawPoints(A,B,C) \tkzDrawCircles(A,C A,B) \tkzDrawLines(A,B' E,t1 E,t2 t1,t2 A,E) \tkzDrawPoints(A,B,C,E,t1,t2,H,B',E') \tkzLabelPoints(A,B,C,E,t1,t2,B',E') \tkzLabelPoints[above](C') \end{tikzpicture} \end{minipage} % subsection inversion (end) \subsection{Gergonne point} % (fold) \label{sub:gergonne_point} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} \begin{tkzelements} z.a = point: new(1,0) z.b = point: new(6,2) z.c = point: new(2,5) T = triangle : new (z.a,z.b,z.c) z.g = T : gergonne_point () z.i = T.incenter z.ta,z.tb,z.tc = get_points (T : intouch ()) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(a,b,c) \tkzDrawPoints(a,b,c,g) \tkzLabelPoints(a,b,c) \tkzDrawSegments (a,ta b,tb c,tc) \tkzDrawCircle(i,ta) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{tkzelements} z.a = point: new(1,0) z.b = point: new(6,2) z.c = point: new(2,5) T = triangle : new (z.a,z.b,z.c) z.g = T : gergonne_point () z.i = T.incenter z.ta,z.tb,z.tc = get_points (T : intouch ()) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(a,b,c) \tkzDrawPoints(a,b,c,g) \tkzLabelPoints(a,b,c) \tkzDrawSegments (a,ta b,tb c,tc) \tkzDrawCircle(i,ta) \end{tikzpicture} \end{minipage} % subsection gergonne_point (end) \subsection{Antiparallel through Lemoine point} % (fold) \label{sub:antiparallel_through_lemoine_point} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.a = point: new (0,0) z.b = point: new (5,0) z.c = point: new (1,4) T = triangle: new (z.a,z.b,z.c) z.L = T : lemoine_point () L.anti = T : antiparallel (z.L,0) z.x_0,z.x_1 = get_points (L.anti) L.anti = T : antiparallel (z.L,1) z.y_0,z.y_1 = get_points (L.anti) L.anti = T : antiparallel (z.L,2) z.z_0,z.z_1 = get_points (L.anti) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(a,b,c) \tkzDrawPoints(a,b,c,L,x_0,x_1,y_0,y_1,z_0,z_1) \tkzLabelPoints(a,b) \tkzLabelPoints[above](L,c) \tkzDrawSegments(x_0,x_1 y_0,y_1 z_0,z_1) \tkzDrawCircle(L,x_0) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tkzelements} z.a = point: new (0,0) z.b = point: new (5,0) z.c = point: new (1,4) T = triangle: new (z.a,z.b,z.c) z.L = T : lemoine_point () L.anti = T : antiparallel (z.L,0) z.x_0,z.x_1 = get_points (L.anti) L.anti = T : antiparallel (z.L,1) z.y_0,z.y_1 = get_points (L.anti) L.anti = T : antiparallel (z.L,2) z.z_0,z.z_1 = get_points (L.anti) \end{tkzelements} \hspace*{\fill} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygons(a,b,c) \tkzDrawPoints(a,b,c,L,x_0,x_1,y_0,y_1,z_0,z_1) \tkzLabelPoints(a,b) \tkzLabelPoints[above](L,c) \tkzDrawSegments(x_0,x_1 y_0,y_1 z_0,z_1) \tkzDrawCircle(L,x_0) \end{tikzpicture} \end{minipage} % subsection antiparallel_through_lemoine_point (end) \subsection{Soddy circle without function} % (fold) \label{sub:soddy} \begin{Verbatim} \begin{tkzelements} z.A = point : new ( 0 , 0 ) z.B = point : new ( 5 , 0 ) z.C = point : new ( 0.5 , 4 ) T.ABC = triangle : new ( z.A,z.B,z.C ) z.I = T.ABC.incenter z.E,z.F,z.G = T.ABC : projection (z.I) C.ins = circle : new (z.I,z.E) T.orthic = T.ABC : orthic () z.Ha,z.Hb,z.Hc = get_points (T.orthic) C.CF = circle : new ( z.C , z.F ) C.AG = circle : new ( z.A , z.G ) C.BE = circle : new ( z.B , z.E ) L.Ah = line : new ( z.A , z.Ha ) L.Bh = line : new ( z.B , z.Hb ) L.Ch = line : new ( z.C , z.Hc ) z.X,z.Xp = intersection (L.Ah,C.AG) z.Y,z.Yp = intersection (L.Bh,C.BE) z.Z,z.Zp = intersection (L.Ch,C.CF) L.XpE = line : new (z.Xp,z.E) L.YpF = line : new (z.Yp,z.F) L.ZpG = line : new (z.Zp,z.G) z.S = intersection (L.XpE,L.YpF) z.Xi = intersection(L.XpE,C.AG) z.Yi = intersection(L.YpF,C.BE) _,z.Zi = intersection(L.ZpG,C.CF) z.S = triangle : new (z.Xi,z.Yi,z.Zi).circumcenter C.soddy_int = circle : new (z.S,z.Xi) C.soddy_ext = C.ins : inversion (C.soddy_int) z.w = C.soddy_ext.center z.s = C.soddy_ext.through z.Xip,z.Yip,z.Zip = C.ins : inversion (z.Xi,z.Yi,z.Zi) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawPoints(A,B,C,E,F,G,Ha,Hb,Hc,X,Y,Z,X',Y',Z',Xi,Yi,Zi,I) \tkzDrawPoints(Xi',Yi',Zi',S) \tkzLabelPoints(A,B,C,E,F,G,X,Y,Z,X',Y',Z') \tkzDrawCircles(A,G B,E C,F I,E S,Xi w,s) \tkzDrawLines(X',Ha Y',Hb Z',Hc) \tkzDrawLines(X',E Y',F Z',G) \end{tikzpicture} \end{Verbatim} \begin{tkzelements} z.A = point : new ( 0 , 0 ) z.B = point : new ( 5 , 0 ) z.C = point : new ( 0.5 , 4 ) T.ABC = triangle : new ( z.A,z.B,z.C ) z.I = T.ABC.incenter z.E,z.F,z.G = T.ABC : projection (z.I) C.ins = circle : new (z.I,z.E) T.orthic = T.ABC : orthic () z.Ha,z.Hb,z.Hc = get_points (T.orthic) C.CF = circle : new ( z.C , z.F ) C.AG = circle : new ( z.A , z.G ) C.BE = circle : new ( z.B , z.E ) L.Ah = line : new ( z.A , z.Ha ) L.Bh = line : new ( z.B , z.Hb ) L.Ch = line : new ( z.C , z.Hc ) z.X,z.Xp = intersection (L.Ah,C.AG) z.Y,z.Yp = intersection (L.Bh,C.BE) z.Z,z.Zp = intersection (L.Ch,C.CF) L.XpE = line : new (z.Xp,z.E) L.YpF = line : new (z.Yp,z.F) L.ZpG = line : new (z.Zp,z.G) z.S = intersection (L.XpE,L.YpF) z.Xi = intersection(L.XpE,C.AG) z.Yi = intersection(L.YpF,C.BE) _,z.Zi = intersection(L.ZpG,C.CF) z.S = triangle : new (z.Xi,z.Yi,z.Zi).circumcenter C.soddy_int = circle : new (z.S,z.Xi) C.soddy_ext = C.ins : inversion (C.soddy_int) z.w = C.soddy_ext.center z.s = C.soddy_ext.through z.Xip,z.Yip,z.Zip = C.ins : inversion (z.Xi,z.Yi,z.Zi) \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawPoints(A,B,C,E,F,G,Ha,Hb,Hc,X,Y,Z,X',Y',Z',Xi,Yi,Zi,I) \tkzDrawPoints(Xi',Yi',Zi',S) \tkzLabelPoints(A,B,C,E,F,G,X,Y,Z,X',Y',Z') \tkzDrawCircles(A,G B,E C,F I,E S,Xi w,s) \tkzDrawLines(X',Ha Y',Hb Z',Hc) \tkzDrawLines(X',E Y',F Z',G) \end{tikzpicture} % subsection soddy (end) \subsection{Soddy circle with function} % (fold) \label{sub:soddy_circle_with_function} \begin{tkzelements} z.A = point : new ( 0 , 0 ) z.B = point : new ( 5 , 0 ) z.C = point : new (4 , 4 ) T.ABC = triangle : new ( z.A,z.B,z.C ) z.I = T.ABC.incenter z.E,z.F,z.G = T.ABC : projection (z.I) T.orthic = T.ABC : orthic () z.Ha,z.Hb,z.Hc = get_points (T.orthic) C.ins = circle : new (z.I,z.E) z.s,z.xi,z.yi,z.zi = T.ABC : soddy_center () C.soddy_int = circle : new (z.s,z.xi) C.soddy_ext = C.ins : inversion (C.soddy_int) z.w = C.soddy_ext.center z.t = C.soddy_ext.through z.Xip,z.Yip,z.Zip = C.ins : inversion (z.xi,z.yi,z.zi) \end{tkzelements} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point : new ( 0 , 0 ) z.B = point : new ( 5 , 0 ) z.C = point : new (4 , 4 ) T.ABC = triangle : new ( z.A,z.B,z.C ) z.I = T.ABC.incenter z.E,z.F,z.G = T.ABC : projection (z.I) T.orthic = T.ABC : orthic () z.Ha,z.Hb,z.Hc = get_points (T.orthic) C.ins = circle : new (z.I,z.E) z.s,z.xi,z.yi, z.zi = T.ABC : soddy_center () C.soddy_int = circle : new (z.s,z.xi) C.soddy_ext = C.ins : inversion (C.soddy_int) z.w = C.soddy_ext.center z.t = C.soddy_ext.through z.Xip,z.Yip, z.Zip = C.ins : inversion (z.xi,z.yi,z.zi) \end{tkzelements} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tikzpicture}[scale=.6] \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawCircles(A,G B,E C,F I,E s,xi w,t) \tkzDrawPoints(A,B,C,E,F,G,s,w,xi,t) \tkzLabelPoints(A,B,C) \tkzDrawPoints(A,B,C,E,F,G,Ha,Hb,Hc,xi,yi,zi,I) \tkzDrawPoints(Xi',Yi',Zi') \tkzLabelPoints(A,B,C,E,F,G) \end{tikzpicture} \end{minipage} \begin{Verbatim} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) \tkzDrawCircles(A,G B,E C,F I,E s,xi w,t) \tkzDrawPoints(A,B,C,E,F,G,s,w,xi,t) \tkzLabelPoints(A,B,C) \tkzDrawPoints(A,B,C,E,F,G,Ha,Hb,Hc,xi,yi,zi,I) \tkzDrawPoints(Xi',Yi',Zi') \tkzLabelPoints(A,B,C,E,F,G) \end{tikzpicture} \end{Verbatim} % subsection soddy_circle_with_function (end) \subsection{Pappus chain} % (fold) \label{sub:pappus_chain} \begin{tkzelements} scale =.75 xC,nc = 10,16 xB = xC/tkzphi xD = (xC*xC)/xB xJ = (xC+xD)/2 r = xD-xJ z.A = point : new ( 0 , 0 ) z.B = point : new ( xB , 0) z.C = point : new ( xC , 0) L.AC = line : new (z.A,z.C) z.i = L.AC.mid L.AB = line:new (z.A,z.B) z.j = L.AB.mid z.D = point : new ( xD , 0) C.AC = circle: new (z.A,z.C) for i = -nc,nc do z["J"..i] = point: new (xJ,2*r*i) z["H"..i] = point: new (xJ,2*r*i-r) z["J"..i.."p"], z["H"..i.."p"] = C.AC : inversion (z["J"..i],z["H"..i]) L.AJ = line : new (z.A,z["J"..i]) C.JH = circle: new ( z["J"..i] , z["H"..i]) z["S"..i], z["T"..i] = intersection (L.AJ,C.JH) z["S"..i.."p"], z["T"..i.."p"] = C.AC : inversion (z["S"..i],z["T"..i]) L.SpTp = line:new ( z["S"..i.."p"], z["T"..i.."p"]) z["I"..i] = L.SpTp.mid end \end{tkzelements} \begin{Verbatim} \begin{tkzelements} xC,nc = 10,16 xB = xC/tkzphi xD = (xC*xC)/xB xJ = (xC+xD)/2 r = xD-xJ z.A = point : new ( 0 , 0 ) z.B = point : new ( xB , 0) z.C = point : new ( xC , 0) L.AC = line : new (z.A,z.C) z.i = L.AC.mid L.AB = line:new (z.A,z.B) z.j = L.AB.mid z.D = point : new ( xD , 0) C.AC = circle: new (z.A,z.C) for i = -nc,nc do z["J"..i] = point: new (xJ,2*r*i) z["H"..i] = point: new (xJ,2*r*i-r) z["J"..i.."p"], z["H"..i.."p"] = C.AC : inversion (z["J"..i],z["H"..i]) L.AJ = line : new (z.A,z["J"..i]) C.JH = circle: new ( z["J"..i] , z["H"..i]) z["S"..i], z["T"..i] = intersection (L.AJ,C.JH) z["S"..i.."p"], z["T"..i.."p"] = C.AC : inversion (z["S"..i],z["T"..i]) L.SpTp = line:new ( z["S"..i.."p"], z["T"..i.."p"]) z["I"..i] = L.SpTp.mid end \end{tkzelements} \end{Verbatim} \begin{minipage}{.5\textwidth} \begin{Verbatim} \def\nc{\tkzUseLua{nc}} \begin{tikzpicture}[ultra thin] \tkzGetNodes \tkzDrawCircle[fill=teal!20](i,C) \tkzDrawCircle[fill=PineGreen!60](j,B) \foreach \i in {-\nc,...,0,...,\nc} { \tkzDrawCircle[fill=teal]({I\i},{S\i'}) } \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \def\nc{\tkzUseLua{nc}} \begin{tikzpicture}[ultra thin] \tkzGetNodes \tkzDrawCircle[fill=teal!20](i,C) \tkzDrawCircle[fill=PineGreen!60](j,B) \foreach \i in {-\nc,...,0,...,\nc} { \tkzDrawCircle[fill=teal]({I\i},{S\i'}) } \end{tikzpicture} \vfill \end{minipage} % subsection pappus_chain (end) \subsection{Three Circles} % (fold) \label{sub:three_circles} \begin{tkzelements} function threecircles(c1,r1,c2,r2,c3,h1,h3,h2) local xk = math.sqrt (r1*r2) local cx = (2*r1*math.sqrt(r2))/(math.sqrt(r1)+math.sqrt(r2)) local cy = (r1*r2)/(math.sqrt(r1)+math.sqrt(r2))^2 z[c2] = point : new ( 2*xk , r2 ) z[h2] = point : new (2*xk,0) z[c1] = point : new (0,r1) z[h1] = point : new (0,0) L.h1h2 = line: new(z[h1],z[h2]) z[c3] = point : new (cx,cy) z[h3] = L.h1h2: projection (z[c3]) end threecircles("A",4,"B",3,"C","E","G","F") \end{tkzelements} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} function threecircles(c1,r1,c2,r2,c3,h1,h3,h2) local xk = math.sqrt (r1*r2) local cx = (2*r1*math.sqrt(r2))/(math.sqrt(r1)+math.sqrt(r2)) local cy = (r1*r2)/(math.sqrt(r1)+math.sqrt(r2))^2 z[c2] = point : new ( 2*xk , r2 ) z[h2] = point : new (2*xk,0) z[c1] = point : new (0,r1) z[h1] = point : new (0,0) L.h1h2 = line: new(z[h1],z[h2]) z[c3] = point : new (cx,cy) z[h3] = L.h1h2: projection (z[c3]) end threecircles("A",4,"B",3,"C","E","G","F") \end{tkzelements} \begin{tikzpicture} \tkzGetNodes \tkzDrawSegment[color = red](E,F) \tkzDrawCircle[orange,fill=orange!20](A,E) \tkzDrawCircle[purple,fill=purple!20](B,F) \tkzDrawCircle[teal,fill=teal!20](C,G) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \begin{tikzpicture}[scale=.5] \tkzGetNodes \tkzDrawSegment[color = red](E,F) \tkzDrawCircle[orange,fill=orange!20](A,E) \tkzDrawCircle[purple,fill=purple!20](B,F) \tkzDrawCircle[teal,fill=teal!20](C,G) \end{tikzpicture} \end{minipage} % subsection three_circles (end) \subsection{pentagons in a golden arbelos} % (fold) \label{sub:golden_arbelos} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (10 , 0) L.AB = line: new ( z.A, z.B) z.C = L.AB : gold_ratio () L.AC = line: new ( z.A, z.C) L.CB = line: new ( z.C, z.B) z.O_0 = L.AB.mid z.O_1 = L.AC.mid z.O_2 = L.CB.mid C.O0B = circle: new ( z.O_0, z.B) C.O1C = circle: new ( z.O_1, z.C) C.O2B = circle: new ( z.O_2, z.B) z.M_0 = C.O1C:external_similitude(C.O2B) L.O0C = line:new(z.O_0,z.C) T.golden = L.O0C : golden () z.L = T.golden.pc L.O0L = line:new(z.O_0,z.L) z.D = intersection (L.O0L,C.O0B) L.DB = line:new(z.D,z.B) _,z.Z = intersection (L.DB,C.O2B) L.DA = line:new(z.D,z.A) z.I = intersection (L.DA,C.O1C) L.O2Z = line:new(z.O_2,z.Z) z.H = intersection (L.O2Z,C.O0B) C.BD = circle:new (z.B,z.D) C.DB = circle:new (z.D,z.B) _,z.G = intersection (C.BD,C.O0B) z.E = intersection (C.DB,C.O0B) C.GB = circle:new (z.G,z.B) _,z.F = intersection (C.GB,C.O0B) k = 1/tkzphi^2 kk = tkzphi z.D_1,z.E_1,z.F_1,z.G_1 = z.B : homothety (k, z.D,z.E,z.F,z.G) z.D_2,z.E_2,z.F_2,z.G_2 = z.M_0 : homothety (kk,z.D_1,z.E_1,z.F_1,z.G_1) \end{tkzelements} \begin{minipage}{.5\textwidth} \begin{Verbatim} \begin{tkzelements} z.A = point: new (0 , 0) z.B = point: new (10 , 0) L.AB = line: new ( z.A, z.B) z.C = L.AB : gold_ratio () L.AC = line: new ( z.A, z.C) L.CB = line: new ( z.C, z.B) z.O_0 = L.AB.mid z.O_1 = L.AC.mid z.O_2 = L.CB.mid C.O0B = circle: new ( z.O_0, z.B) C.O1C = circle: new ( z.O_1, z.C) C.O2B = circle: new ( z.O_2, z.B) z.M_0 = C.O1C:external_similitude(C.O2B) L.O0C = line:new(z.O_0,z.C) T.golden = L.O0C : golden () z.L = T.golden.pc L.O0L = line:new(z.O_0,z.L) z.D = intersection (L.O0L,C.O0B) L.DB = line:new(z.D,z.B) z.Z = intersection (L.DB,C.O2B) L.DA = line:new(z.D,z.A) z.I = intersection (L.DA,C.O1C) L.O2Z = line:new(z.O_2,z.Z) z.H = intersection (L.O2Z,C.O0B) C.BD = circle:new (z.B,z.D) C.DB = circle:new (z.D,z.B) _,z.G = intersection (C.BD,C.O0B) z.E = intersection (C.DB,C.O0B) C.GB = circle:new (z.G,z.B) _,z.F = intersection (C.GB,C.O0B) k = 1/tkzphi^2 kk = tkzphi z.D_1,z.E_1,z.F_1,z.G_1 = z.B : homothety (k, z.D,z.E,z.F,z.G) z.D_2,z.E_2,z.F_2,z.G_2 = z.M_0 : homothety (kk,z.D_1,z.E_1,z.F_1,z.G_1) \end{tkzelements} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} \hfill \begin{tikzpicture}[scale=.7] \tkzGetNodes \tkzDrawPolygon[red](O_2,O_0,I,D,H) \tkzDrawPolygon[blue](B,D_1,E_1,F_1,G_1) \tkzDrawPolygon[green](C,D_2,E_2,F_2,G_2) \tkzDrawPolygon[purple](B,D,E,F,G) \tkzDrawCircles(O_0,B O_1,C O_2,B) \tkzFillPolygon[fill=red!20,opacity=.20](O_2,O_0,I,D,H) \tkzFillPolygon[fill=blue!20,opacity=.20](B,D_1,E_1,F_1,G_1) \tkzFillPolygon[fill=green!60,opacity=.20](C,D_2,E_2,F_2,G_2) \tkzFillPolygon[fill=purple!20,opacity=.20](B,D,E,F,G) \tkzDrawCircles(O_0,B O_1,C O_2,B) \tkzDrawSegments[new](A,B) \tkzDrawPoints(A,B,C,O_0,O_1,O_2,Z,I,H,B,D,E,F) \tkzDrawPoints(D_1,E_1,F_1,G_1) \tkzDrawPoints(D_2,E_2,F_2,G_2) \tkzDrawPoints[red](F_1) \tkzLabelPoints(A,B,C,O_0,O_2) \tkzLabelPoints[below](O_1,G) \tkzLabelPoints[above right](D,H) \tkzLabelPoints[above left](E,E_1,E_2) \tkzLabelPoints[below left](F,F_1,F_2) \tkzLabelPoints(D_1,G_1) \tkzLabelPoints(D_2,G_2) \end{tikzpicture} \end{minipage} \begin{Verbatim} \begin{tikzpicture}[scale=.8] \tkzGetNodes \tkzDrawPolygon[red](O_2,O_0,I,D,H) \tkzDrawPolygon[blue](B,D_1,E_1,F_1,G_1) \tkzDrawPolygon[green](C,D_2,E_2,F_2,G_2) \tkzDrawPolygon[purple](B,D,E,F,G) \tkzDrawCircles(O_0,B O_1,C O_2,B) \tkzFillPolygon[fill=red!20,opacity=.20](O_2,O_0,I,D,H) \tkzFillPolygon[fill=blue!20,opacity=.20](B,D_1,E_1,F_1,G_1) \tkzFillPolygon[fill=green!60,opacity=.20](C,D_2,E_2,F_2,G_2) \tkzFillPolygon[fill=purple!20,opacity=.20](B,D,E,F,G) \tkzDrawCircles(O_0,B O_1,C O_2,B) \tkzDrawSegments[new](A,B) \tkzDrawPoints(A,B,C,O_0,O_1,O_2,Z,I,H,B,D,E,F) \tkzDrawPoints(D_1,E_1,F_1,G_1) \tkzDrawPoints(D_2,E_2,F_2,G_2) \tkzDrawPoints[red](F_1) \tkzLabelPoints(A,B,C,O_0,O_2) \tkzLabelPoints[below](O_1,G) \tkzLabelPoints[above right](D,H) \tkzLabelPoints[above left](E,E_1,E_2) \tkzLabelPoints[below left](F,F_1,F_2) \tkzLabelPoints(D_1,G_1) \tkzLabelPoints(D_2,G_2) \end{tikzpicture} \end{Verbatim} % subsection golden_arbelos (end)