X₁₃₄₅

EXSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)

X₁₃₄₅ is the exsimilicenter(circumcircle, orthocentroidal circle).

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₃₄₅ (EXSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {55, 2463}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₅₅ (INSIMILICENTER), X₂₄₆₃ (INSIMILICENTER)

Trilinears: \((|OH| - R) \cos A - 4R \cos B \cos C : :\)

Barycentrics: \(a^{4}+a^{2} b^{2}-2 b^{4}+a^{2} c^{2}+4 b^{2} c^{2}-2 c^{4}+a^{2} (a^{2}-b^{2}-c^{2}) J : :\)

Search Value (6, 9, 13): \(-16.934794786887\)

▸ Linear Combinations1 combination

• X₁₃₄₅ = \(6\) X₂ - \((3 + |OH|/R)\) X₃

X₁₃₄₅ lies on the cubics K281, K297, K708, K843.

▸ Incident Central Lines23 lines

{2, 3}, {6, 2575}, {55, 2463}, {56, 2464}, {64, 14374}, {111, 8426}, {112, 8427}, {183, 15164}, {371, 2465}, {372, 2466}, {958, 2467}, {1340, 2469}, {1341, 2470}, {1342, 2471}, {1343, 2472}, {1351, 8115}, {1376, 2468}, {1689, 2016}, {1690, 2015}, {2574, 11472}, {5640, 24650}, {5968, 16071}, {6090, 8116}

▸ Functional Relations7 relations

• X₁₃₄₅ = external center of similitude of circumcircle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

• X₁₃₄₅ = orthocentroidal circle inverse of X₁₃₁₂

• X₁₃₄₅ = X₁₁₄₇₂-line conjugate of X₂₅₇₄

• X₁₃₄₅ = X₁₃₄₄-vertex conjugate of X₁₁₁₈₁

• X₁₃₄₅ = crossdifference of every pair of points on line {647, 2574}

• X₁₃₄₅ = homothetic center of orthocentroidal triangle and circumcevian triangle of X₂₅₇₅

• X₁₃₄₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 4, 1312), (2, 1114, 3), (2, 1347, 5094), (2, 1995, 1344), (2, 2552, 1346), (3, 381, 1344), (3, 3830, 15154), (4, 378, 1344), (4, 1114, 25), (4, 1347, 381), (4, 10736, 3830), (4, 14710, 3), (5, 6644, 1344), (5, 20479, 3), (22, 5169, 1344), (23, 10720, 15154), (24, 7577, 1344), (25, 5094, 1344), (458, 4230, 1344), (868, 3148, 1344), (1114, 10720, 23), (1346, 2552, 381), (1347, 14710, 378), (2070, 7579, 1344), (3091, 14710, 3516), (3091, 15078, 1344), (3526, 21308, 1344), (7418, 13860, 1344), (7485, 7533, 1344), (14808, 15159, 20063)

Notes

As a point on the Euler line, X₁₃₄₅ has Shinagawa coefficients \((R - |OH|,3R + |OH|)\).