X2334

X2-ISOCONJUGATE OF X1449

X2334 = X1191-cross conjugate of X56
< 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) { 8, 86 }: Central Line L(8, 86) { 42, 56 }: Central Line L(42, 56) { 55, 58 }: Central Line L(55, 58) { 1, 210 }: Central Line L(1, 210) { 65, 269 }: Central Line L(65, 269) { 106, 386 }: Central Line L(106, 386) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1191 = 2nd SARAGOSSA POINT OF X(56) (0.6211, 0.2922) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_8 = NAGEL POINT (0.4694, 0.1333) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_1 = INCENTER (0.5714, 0.2333) X_210 = -CEVA CONJUGATE OF X(37) (0.4925, 0.1466) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_2334 = -ISOCONJUGATE OF X(1449) (0.5898, 0.2535) A B C X1191 X56 X8 X86 X42 X55 X58 X1 X210 X65 X269 X106 X386 X2334 (a, b, c) = (5, 6, 7) | AB = 1

X2334 (X2-ISOCONJUGATE OF X1449) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\), where f(a,b,c) is as in X1449
Barycentrics: \(a^{2}/f(a,b,c) : b^{2}/f(b,c,a): c^{2}/f(c,a,b)\)
Search Value (6, 9, 13): \(1.226024821148\)
▸ Incident Central Lines14 lines

{1, 210}, {3, 2163}, {6, 1334}, {8, 86}, {34, 1824}, {42, 56}, {55, 58}, {65, 269}, {106, 386}, {145, 1220}, {607, 1474}, {998, 2098}, {1413, 2357}, {1449, 2297}

▸ Functional Relations2 relations

Notes

Let A'B'C' = circumcevian triangle of X1, and let

  • Oa = center of A-mixtilinear incircle

  • Ta = touchpoint of A-mixtilinear incircle and circumcircle

  • A1 = circumcenter of A’OaTa, and define B1 and C1 cyclically. Then AA1, BB1, CC1 concur in X2334. (Angel Montesdeoca, November 29, 2018)

Let DEF be the cevian triangle of X8, and let A5 is the QA-P5 center (Isotomic Center) of AEFX8 (see Chris van Tienhoven, Quadrangle points and other objects). Let B5 be the QA-P5 center of BFDX8 and C5 the QA-P5 center of CDEX8. Then the triangles ABC and A5B5C5 are perspective, and X2334 is their perspector. (Angel Montesdeoca, March 20, 2019)

In the plane of a triangle ABFC, let

  • DEF = cevian triangle of X1;

  • Γ = circumcircle;

  • Aa = pole of EF wrt Γ

  • T1 and T2 = lines through Aa tangent to BC;

  • Ab and Ac = T1∩BC and T2∩BC;

  • Define Bc and Ca cyclically, and define Ba and Cb cyclically.

  • The six points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic, 𝒞, given by the barycentric equation

  • b^2 c^2x^2+a^2 c^2y^2+a^2 b^2z^2+a^2 (a^2+b^2+c^2+2 a (b+c))yz+b^2 (a^2+2 a b+(b+c)^2)zx+c^2 (a^2+2 a c+(b+c)^2)xy= 0. The polars of Aa, Bb, Cc with respect to 𝒞 form a perspective triangle to ABC, and the perspector is X2334. (Angel Montesdeoca, November 27, 2022)