X63

ISOGONAL CONJUGATE OF X19

X63 is the isogonal conjugate of X19.
< 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) { 2, 7 }: Centroid–Gergonne Line { 1, 21 }: Central Line L(1, 21) { 8, 20 }: Central Line L(8, 20) { 19, 27 }: Central Line L(19, 27) { 10, 46 }: Central Line L(10, 46) { 3, 72 }: Central Line L(3, 72) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_19 = CLAWSON POINT (0.6246, 0.4244) X_2 = CENTROID (0.5374, 0.2000) X_7 = GERGONNE POINT (0.5903, 0.2769) X_1 = INCENTER (0.5714, 0.2333) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_8 = NAGEL POINT (0.4694, 0.1333) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_27 = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER (0.6182, 0.4112) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4367, 0.0733) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) A B C X19 X2 X7 X1 X21 X8 X20 X27 X10 X46 X3 X72 X63 (a, b, c) = (5, 6, 7) | AB = 1

X63 (ISOGONAL CONJUGATE OF X19) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cot A : \cot B : \cot C\)
Barycentrics: \(\cos A : \cos B : \cos C\)
Search Value (6, 9, 13): \(5.874900573472\)
▸ Alternate Coordinates7 forms

  • Trilinears: \(b^{2} + c^{2} - a^{2} : c^{2} + a^{2} - b^{2} : a^{2} + b^{2} - c^{2}\)

  • Trilinears: \(S_{A} : S_{B} : S_{C}\)

  • Trilinears: \(\csc A - \tan(A/2) : :\)

  • Trilinears: \(\csc A - \cot(A/2) : :\)

  • Trilinears: \(\tan(A/2) - \cot(A/2) : :\)

  • Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = directed distance from A to the orthic axis

  • Trilinears: \(2 \csc 2A - \tan A : :\)

▸ Linear Combinations2 combinations

  • X63 = \((r + 2R)\) X1 - \(3R\) X2 - \(2r\) X3 (Peter Moses, April 2, 2013)

  • X63 = \((\cos A)\) A + \((\cos B)\) B + \((\cos C)\) C, where A, B, C denote angles and A, B, C denote vertices

▸ Incident Central Lines45 lines

{1, 21}, {2, 7}, {3, 72}, {6, 2221}, {8, 20}, {10, 46}, {19, 27}, {33, 1013}, {36, 997}, {37, 940}, {48, 326}, {55, 518}, {56, 960}, {65, 958}, {69, 71}, {77, 219}, {91, 921}, {100, 103}, {162, 204}, {169, 379}, {171, 612}, {190, 312}, {194, 239}, {201, 603}, {210, 1004}, {212, 1040}, {213, 980}, {220, 241}, {223, 651}, {238, 614}, {240, 1096}, {244, 748}, {304, 1102}, {318, 412}, {354, 1001}, {392, 999}, {404, 936}, {405, 942}, {452, 938}, {484, 535}, {517, 956}, {544, 1018}, {561, 799}, {654, 918}, {750, 756}

▸ Functional Relations41 relations

Notes

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius \(\sqrt{r_a^{2} + s^{2}}\), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is perspective to the excentral triangle at X63. (Randy Hutson, February 10, 2016)

Let A'B'C' be the 2nd Brocard triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X63. (Randy Hutson, February 10, 2016)

Let A'B'C' be the hexyl triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X63. (Randy Hutson, February 10, 2016)

Let A'B'C' be the side-triangle of ABC and hexyl triangle. Let A″ be the {B,C}-harmonic conjugate of A', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X63. (Randy Hutson, February 10, 2016)

Let JaJbJc be the excentral triangle. Let A″ be the isotomic conjugate, wrt triangle JaBC, of X1. Define B″ and C″ cyclically. The lines JaA″, JbB″, JcC″ concur in X63. (Randy Hutson, July 31 2018)

X63 is the {X9,X57}-harmonic conjugate of X2.