X3431

X4-VERTEX CONJUGATE OF X6

X3431 = isogonal conjugate of X381
< 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, 186 }: Central Line L(6, 186) { 2, 265 }: Central Line L(2, 265) { 3, 323 }: Central Line L(3, 323) { 68, 631 }: Central Line L(68, 631) { 248, 574 }: Central Line L(248, 574) { 4, 1495 }: Central Line L(4, 1495) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_566 = HARMONIC OF X(50) (0.5516, 0.1872) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_265 = REFLECTION OF X(3) IN X(125) (0.8606, -0.0500) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_323 = ISOTOMIC CONJUGATE OF X(94) (-0.8839, -1.3275) X_68 = PRASOLOV POINT (-0.0535, -0.0055) X_631 = 3/5*OG (0.5224, 0.1608) X_248 = -LINE CONJUGATE OF X(132) (0.8229, -0.0790) X_574 = HARMONIC OF X(187) (0.5539, 0.1910) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1495 = CROSSSUM OF X(2) AND X(30) (0.8194, 1.4855) X_3431 = -VERTEX CONJUGATE OF X(6) (0.5702, 0.1746) A B C X566 X2 X6 X186 X265 X3 X323 X68 X631 X248 X574 X4 X1495 X3431 (a, b, c) = (5, 6, 7) | AB = 1

X3431 (X4-VERTEX CONJUGATE OF X6) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a/[a^{2}vwyz - ux(bw + cv)(bz + cy)]\), where \(u : v : w\) = X4 and \(x : y : z\) = X6
Barycentrics: \(a^{2}/[a^{4} + a^{2}(b^{2} + c^{2}) - 2(b^{2} - c^{2})^{2}] : : (E. Danneels)\)
Search Value (6, 9, 13): \(5.609487026197\)
▸ Alternate Coordinates1 form

  • Barycentrics: \((S_{B} + S_{C})/(S^{2} + 3 S_{B} S_{C}) : :\)

▸ Incident Central Lines6 lines

{2, 265}, {3, 323}, {4, 1495}, {6, 186}, {68, 631}, {248, 574}

▸ Functional Relations5 relations

Notes

Let A'B'C' be the orthocentroidal triangle. X3431 is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)

In the plane of a triangle ABC, let

  • Oa = circle with diameter BC, and define Ob and Oc cyclically;

  • A' = reflection of A in Oa, and define B' and C' cyclically;

  • Pa = polar of A' with respect to Oa, and define Pb and Pc cyclically;

  • A″ = Pc∩Pb, and define B″ and C″ cyclically.

The triangle A″B″C″ is perspective to ABC, and the perspector is X3431. (Dasari Naga Vijay Krishna, June 8, 2021)