X2050

{X3,X5}-HARMONIC CONJUGATE OF X2049

X2050 is the {X3,X5}-harmonic conjugate of X2049.
< 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) { 3, 5 }: Euler Line { 11, 1460 }: Central Line L(11, 1460) { 171, 1699 }: Central Line L(171, 1699) { 1482, 1999 }: Central Line L(1482, 1999) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2049 = EULER LINE INTERCEPT OF LINE X(6)X(10) (0.5382, 0.2021) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_1460 = -BETH CONJUGATE OF X(612) (0.6249, 0.3315) X_171 = {X(2), X(31)}-HARMONIC CONJUGATE OF X(238) (0.5819, 0.2525) X_1699 = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE) (0.5986, 0.3416) X_1482 = REFLECTION OF CIRCUMCENTER IN INCENTER (0.6429, 0.3645) X_1999 = ORTHOCORRESPONDENT OF X(10) (0.6884, 0.3479) X_2050 = {X(3),X(5)}-HARMONIC CONJUGATE OF X(2049) (0.6055, 0.3782) A B C X2049 X3 X5 X11 X1460 X171 X1699 X1482 X1999 X2050 (a, b, c) = (5, 6, 7) | AB = 1

X2050 ({X3,X5}-HARMONIC CONJUGATE OF X2049) 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 \(f(A,B,C) = (s^{2} - r^{2})\cos A - (s^{2} + r^{2})\cos(B - C)\), where r = inradius, s = semiperimeter
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-5.328603145420\)
▸ Incident Central Lines4 lines

{2, 3}, {11, 1460}, {171, 1699}, {1482, 1999}

Notes

As a point on the Euler line, X2050 has Shinagawa coefficients \((r^{2}, s^{2})\).