X215

X1-CEVA CONJUGATE OF X50

X215 is the insimilicenter of the incircle and the sine-triple-angle circle. (Randy Hutson, December 14, 2014)
< 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) { 1, 49 }: Central Line L(1, 49) { 12, 54 }: Central Line L(12, 54) { 11, 110 }: Central Line L(11, 110) { 55, 184 }: Central Line L(55, 184) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_50 = -CEVA CONJUGATE OF X(184) (2.1559, 2.8368) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_54 = KOSNITA POINT (0.5880, 0.2176) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_184 = INVERSE OF X(125) IN THE BROCARD CIRCLE (0.5788, 0.1496) X_215 = -CEVA CONJUGATE OF X(50) (0.4113, 0.3887) A B C X1 X50 X49 X12 X54 X11 X110 X55 X184 X215 (a, b, c) = (5, 6, 7) | AB = 1

X215 (X1-CEVA CONJUGATE OF X50) 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 \(f(A,B,C) = (\sin 3A)(\sin 3B + \sin 3C - \sin 3A)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.577243820289\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(\cos^{2}(3A/2) : \cos^{2}(3B/2) : \cos^{2}(3C/2)\) (M. Iliev, 4/12/07)

  • Trilinears: \(1 + \cos 3A : 1 + \cos 3B : 1+ \cos 3C\) (M. Iliev, 4/12/07)

  • Trilinears: \(a^{3} (a^{2} - b^{2} - c^{2} + b c)^{2} (a - b - c) : :\)

▸ Incident Central Lines4 lines

{1, 49}, {11, 110}, {12, 54}, {55, 184}

▸ Functional Relations1 relation