X3567

DON QUIXOTE POINT

The other two Spanish Points are the Sancho Panza Point X3060 and the Miguel de Cervantes Point X143.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 6, 24 }: Central Line L(6, 24) { 2, 52 }: Central Line L(2, 52) { 4, 51 }: Central Line L(4, 51) { 3, 143 }: Central Line L(3, 143) { 5, 568 }: Central Line L(5, 568) { 186, 578 }: Central Line L(186, 578) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_2 = CENTROID (0.5374, 0.2000) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_4 = ORTHOCENTER (0.6122, 0.3958) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_568 = ORTHOHARMONIC OF X(50) (0.6437, 0.3393) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_578 = ORTHOHARMONIC OF X(216) (0.5933, 0.2561) X_3567 = DON QUIXOTE POINT (0.6087, 0.3032) A B C X6 X24 X2 X52 X4 X51 X3 X143 X5 X568 X186 X578 X3567 (a, b, c) = (5, 6, 7) | AB = 1

X3567 (DON QUIXOTE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[a^{6}(b^{2} + c^{2}) - a^{4}(3b^{4} + b^{2}c^{2} + 3c^{4}) + 3a^{2}(b^{2} + c^{2})(b^{2} - c^{2})^{2} - (b^{4} -b^{2}c^{2} + c^{4})(b^{2} - c^{2})^{2}]\)
Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}[2S^{4} + S_{B}S_{C}(2S^{2} - S_{B}S_{C} - (1/2)(^{2} + b^{2} + c^{2})S_{A}S_{B}S_{C}\)
Search Value (6, 9, 13): \(1.280074581434\)
▸ Incident Central Lines16 lines

{2, 52}, {3, 143}, {4, 51}, {5, 568}, {6, 24}, {25, 1614}, {74, 1112}, {140, 2979}, {155, 1995}, {184, 1199}, {186, 578}, {511, 631}, {567, 1658}, {576, 1092}, {1147, 1994}, {1154, 1656}

▸ Functional Relations5 relations

  • X3567 = X1698-of-the-orthic-triangle, if ABC is acute. (See also X3060.)

  • X3567 = crosssum of X3 and X1656

  • X3567 = homothetic center of orthocentroidal triangle and X4-adjunct anti-altimedial triangle

  • X3567 = homothetic center of X5-altimedial and X2-adjunct anti-altimedial triangles

  • X3567 = insimilicenter of the circumcircle and the nine-point circle of the orthic triangle. (The exsimilicenter is X3060.)

Notes

This point is the first of three Spanish Points developed by Antreas P. Hatzipolakis and Javier Garcia Capitan in 2009; see