X₂₅₄

X₃-CROSS CONJUGATE OF X₄

X₂₅₄ = isogonal conjugate of X₁₅₅

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

X₂₅₄ (X₃-CROSS CONJUGATE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 155}, {68, 136}

• Related Centers: X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₅₅ (EIGENCENTER OF ORTHIC TRIANGLE), X₆₈ (PRASOLOV POINT), X₁₃₆ (INTERSECTION OF LINE X)

Trilinears: \((\sec A)/(\cos^{2} B + \cos^{2} C - \cos^{2} A) : :\)

Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\tan A)/(\cos^{2} B + \cos^{2} C - \cos^{2} A)\)

Search Value (6, 9, 13): \(7.755594037951\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(1/((a^{2} - b^{2} - c^{2}) (a^{6} - 3 a^{4} (b^{2} + c^{2}) + a^{2} (3 b^{4} - 2 b^{2} c^{2} + 3 c^{4}) - (b^{2} - c^{2})^{2} (b^{2} + c^{2}))) : :\)

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{6} - a^{4} b^{2} - a^{2} b^{4} + b^{6} - 3 a^{4} c^{2} + 2 a^{2} b^{2} c^{2} - 3 b^{4} c^{2} + 3 a^{2} c^{4} + 3 b^{2} c^{4} - c^{6}) \cdot (a^{6} - 3 a^{4} b^{2} + 3 a^{2} b^{4} - b^{6} - a^{4} c^{2} + 2 a^{2} b^{2} c^{2} + 3 b^{4} c^{2} - a^{2} c^{4} - 3 b^{2} c^{4} + c^{6}) : :\)

X₂₅₄ lies on the following curves: the circumconics {A, B, C,G,X₂₄}}, {A, B, C,O,X₁₅₅}}, {A, B, C,H,X₉₃}}; the cubics K006, K045, K163, K364, K491, K519, K617, K919; Q066;.

▸ Incident Central Lines26 lines

{2, 847}, {3, 33495}, {4, 155}, {20, 1300}, {24, 393}, {46, 225}, {68, 136}, {186, 18126}, {264, 3541}, {378, 1217}, {403, 6526}, {487, 24244}, {488, 24243}, {1092, 8754}, {1093, 3542}, {1147, 14593}, {1179, 7487}, {1594, 18855}, {1826, 17857}, {2970, 3548}, {3564, 20422}, {6344, 7505}, {8801, 15559}, {8883, 8884}, {8889, 18854}, {18560, 18850}

▸ Functional Relations17 relations

• X₂₅₄ = reflection of X₃₃₄₉₅ in X₃

• X₂₅₄ = isogonal conjugate of X₁₅₅

• X₂₅₄ = isotomic conjugate of isogonal conjugate of X₃₉₁₀₉

• X₂₅₄ = anticomplement of X₃₄₈₅₃

• X₂₅₄ = polar conjugate of X₆₅₁₅

• X₂₅₄ = cyclocevian conjugate of X₁₃₅₇₉

• X₂₅₄ = isogonal conjugate of the anticomplement of X₁₂₃₅₉

• X₂₅₄ = isogonal conjugate of the complement of X₁₁₄₁₁

• X₂₅₄ = isotomic conjugate of the anticomplement of X₂₁₆₅

• X₂₅₄ = polar conjugate of the isotomic conjugate of X₆₅₀₄

• X₂₅₄ = X_i-cross conjugate of X_j for these (i,j): {3, 4}, {2165, 2}

• X₂₅₄ = X_i-isoconjugate of X_j for these (i,j): {1, 155}, {3, 920}, {19, 6503}, {48, 6515}, {63, 1609}, {184, 33808}, {255, 3542}, {454, 921}, {1725, 15478}

• X₂₅₄ = cevapoint of X_i and X_j for these (i,j): {3, 15316}, {4, 3147}, {136, 523}, {647, 8754}

• X₂₅₄ = trilinear pole of line {924, 2501}

• X₂₅₄ = barycentric product X_i*X_j for these {i,j}: {4, 6504}, {92, 921}, {275, 8800}, {2052, 15316}, {2986, 16172}, {11547, 32132}, {13398, 14618}

• X₂₅₄ = barycentric quotient X_i/X_j for these {i,j}: {3, 6503}, {4, 6515}, {6, 155}, {19, 920}, {25, 1609}, {92, 33808}, {393, 3542}, {921, 63}, {1609, 454}, {6504, 69}, {8800, 343}, {8882, 8883}, {13398, 4558}, {14910, 15478}, {15316, 394}, {16172, 3580}, {16310, 27087}

• X₂₅₄ = {X₈₈₀₀,X₁₅₃₁₆}-harmonic conjugate of X₆₅₀₄