X2049

EULER LINE INTERCEPT OF LINE X6X_10

< 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, 3 }: Euler Line { 6, 10 }: Central Line L(6, 10) { 12, 1460 }: Central Line L(12, 1460) { 171, 1698 }: Central Line L(171, 1698) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) 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_1698 = INSIMILICENTER(BEVAN CIRCLE, NINE-POINTCIRCLE) (0.5306, 0.1933) X_2049 = EULER LINE INTERCEPT OF LINE X(6)X(10) (0.5382, 0.2021) A B C X2 X3 X4 X6 X10 X12 X1460 X171 X1698 X2049 (a, b, c) = (5, 6, 7) | AB = 1

X2049 (EULER LINE INTERCEPT OF LINE X6X_10) 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): \(2.589204439474\)
▸ Incident Central Lines4 lines

{2, 3}, {6, 10}, {12, 1460}, {171, 1698}

Notes

As a point on the Euler line, X2049 has Shinagawa coefficients \((s^{2},r^{2})\).