X195

X5-CEVA CONJUGATE OF X3

X195 = X5-Ceva conjugate of X3
< 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, 17 }: Central Line L(6, 17) { 3, 54 }: Central Line L(3, 54) { 49, 52 }: Central Line L(49, 52) { 110, 143 }: Central Line L(110, 143) { 4, 399 }: Central Line L(4, 399) { 140, 323 }: Central Line L(140, 323) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_54 = KOSNITA POINT (0.5880, 0.2176) X_49 = CENTER OF SINE-TRIPLE-ANGLE CIRCLE (0.6275, 0.1789) X_52 = ORTHOCENTER OF ORTHIC TRIANGLE (0.7155, 0.4580) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_4 = ORTHOCENTER (0.6122, 0.3958) X_399 = PARRY REFLECTION POINT (0.0032, 0.9936) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_323 = ISOTOMIC CONJUGATE OF X(94) (-0.8839, -1.3275) X_195 = -CEVA CONJUGATE OF X(3) (0.6761, 0.3331) A B C X5 X3 X6 X17 X54 X49 X52 X110 X143 X4 X399 X140 X323 X195 (a, b, c) = (5, 6, 7) | AB = 1

X195 (X5-CEVA CONJUGATE OF X3) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((\cos A)(v + w - u) : :\), where \(u = u(A,B,C) = \cos A \cos(B - A) \cos(C - A)\)
Barycentrics: \(4 \cos 2A + \cot^{2} A - \cot A \cot \omega : :(\) (M. Iliev, 5/13/07)
Search Value (6, 9, 13): \(-19.157880999134\)
▸ Alternate Coordinates1 form

  • Trilinears: \(a[a^{8} + b^{8} + c^{8} - 4a^{6}(b^{2} + c^{2}) + a^{4}(6b^{4} + 6c^{4} + 5b^{2}c^{2}) - a^{2}(4b^{6} + 4c^{6} - b^{4}c^{2} - b^{2}c^{4}) - 2b^{2}c^{2}(b^{4} + c^{4} - b^{2}c^{2})] : :\)

X195 lies on the Napoleon cubic.

▸ Incident Central Lines12 lines

{1, 3467}, {3, 54}, {4, 399}, {5, 3459}, {6, 17}, {49, 52}, {110, 143}, {140, 323}, {155, 381}, {382, 1498}, {2121, 3462}, {3461, 3468}

▸ Functional Relations12 relations

Notes

Let A' be the isogonal conjugate of the A-vertex of the outer Napoleon triangle, and define B' and C' cyclically. Let A″ be the isogonal conjugate of the A-vertex of the inner Napoleon triangle, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X195. (Randy Hutson, November 18, 2015)

The Napoleon axis and Napoleon-Feuerbach cubic K005 meet in three points: X17, X18, and X195. (Randy Hutson, November 18, 2015)

A construction of X195 is given by Antreas Hatipolakis and Angel Montesdeoca at 24180.