X₁₃₄₄

INSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)

X₁₃₄₄ is the insimilicenter(circumcircle, orthocentroidal circle).

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

X₁₃₄₄ (INSIMILICENTER(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, 2464}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₅₅ (INSIMILICENTER), X₂₄₆₄ (EXSIMILICENTER)

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): \(1.747789575527\)

▸ Linear Combinations1 combination

• X₁₃₄₄ = \(6\) X₂ + \((-3 + |OH|/R)\) X₃

X₁₃₄₄ lies on the cubics K281, K297, K708, K843.

▸ Incident Central Lines23 lines

{2, 3}, {6, 2574}, {55, 2464}, {56, 2463}, {64, 14375}, {111, 8427}, {112, 8426}, {183, 15165}, {371, 2466}, {372, 2465}, {958, 2468}, {1340, 2470}, {1341, 2469}, {1342, 2472}, {1343, 2471}, {1351, 8116}, {1376, 2467}, {1689, 2015}, {1690, 2016}, {2575, 11472}, {5640, 24651}, {5968, 16070}, {6090, 8115}

▸ Functional Relations7 relations

• X₁₃₄₄ = internal center of similitude of circumcircle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

• X₁₃₄₄ = orthocentroidal circle inverse of X₁₃₁₃

• X₁₃₄₄ = X₁₁₄₇₂-line conjugate of X₂₅₇₅

• X₁₃₄₄ = X₁₃₄₅-vertex conjugate of X₁₁₁₈₁

• X₁₃₄₄ = crossdifference of every pair of points on line {647, 2575}

• X₁₃₄₄ = homothetic center of orthocentroidal triangle and circumcevian triangle of X₂₅₇₄

• X₁₃₄₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 4, 1313), (2, 1113, 3), (2, 1346, 5094), (2, 1995, 1345), (2, 2553, 1347), (3, 381, 1345), (3, 3830, 15155), (4, 378, 1345), (4, 1113, 25), (4, 1346, 381), (4, 10737, 3830), (4, 14709, 3), (5, 6644, 1345), (5, 20478, 3), (22, 5169, 1345), (23, 10719, 15155), (24, 7577, 1345), (25, 5094, 1345), (458, 4230, 1345), (868, 3148, 1345), (1113, 10719, 23), (1346, 14709, 378), (1347, 2553, 381), (2070, 7579, 1345), (3091, 14709, 3516), (3091, 15078, 1345), (3526, 21308, 1345), (7418, 13860, 1345), (7485, 7533, 1345), (14807, 15158, 20063)

Notes

As a point on the Euler line, X₁₃₄₄ has Shinagawa coefficients \((R + |OH|,3R - |OH|)\).