X₄₅₉

X₂₅₃-CEVA CONJUGATE OF X₄

X₄₅₉ = X₂₅₃-Ceva conjugate of X₄

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

X₄₅₉ (X₂₅₃-CEVA CONJUGATE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 64}, {2, 253}

• Related Centers: X₂₅₃ (-CROSS CONJUGATE OF X₂), X₄ (ORTHOCENTER), X₆₄ (ISOGONAL CONJUGATE OF X₂₀), X₂ (CENTROID)

Trilinears: \((\sec A)/(\tan B + \tan C - \tan A) : :\)

Barycentrics: \(1/(\tan B \tan C - 2) : :\)

Search Value (6, 9, 13): \(1.857177179962\)

▸ Alternate Coordinates2 forms

• Trilinears: \(|A'A''| : :\), where A'B'C', A″B″C″ are the cevian and anticevian triangles of X₄

• Barycentrics: \(1/(S_{A}(S^{2} - 2S_{B}S_{C})) : :\)

X₄₅₉ lies on the Kiepert hyperbola.

▸ Incident Central Lines20 lines

{2, 253}, {4, 64}, {10, 3176}, {25, 3424}, {69, 801}, {92, 1446}, {96, 3147}, {98, 1301}, {154, 5922}, {196, 226}, {262, 3168}, {297, 2996}, {458, 5395}, {485, 3535}, {486, 3536}, {1075, 3090}, {1131, 1585}, {1132, 1586}, {1503, 3079}, {5485, 5523}

▸ Functional Relations11 relations

• X₄₅₉ = isotomic conjugate of X₃₇₆₆₉

• X₄₅₉ = X₂₅₃-Ceva conjugate of X₄

• X₄₅₉ = X₂₁₅₅-complementary conjugate of X₃₃₅₀

• X₄₅₉ = cevapoint of X_i and X_j for these {i,j}: {6, 3515}, {125, 2501}, {459, 459}, {523, 1562}

• X₄₅₉ = X_i-cross conjugate of X_j for these (i,j): (64, 253), (235, 264), (393, 4), (1562, 523)

• X₄₅₉ = polar conjugate of X₂₀

• X₄₅₉ = X_i-isoconjugate of X_j for these {i,j}: {3, 610}, {20, 48}, {63, 154}, {204, 394}, {219, 1394}, {255, 1249}, {326, 3172}, {577, 1895}, {1101, 1562}, {1259, 3213}, {1790, 3198}, {2193, 5930}

• X₄₅₉ = {X₂,X₂₅₃}-harmonic conjugate of X₁₀₇₃

• X₄₅₉ = trilinear product X_i*X_j for these {i,j}: {4, 2184}, {19, 253}, {64, 92}, {158, 1073}, {264, 2155}, {1301, 1577}, {1880, 5931}}

• X₄₅₉ = barycentric product X_i*X_j for these {i,j}: {4, 253}, {64, 264}, {92, 2184}, {225, 5931}, {850, 1301}, {1073, 2052}, {1969, 2155}

• X₄₅₉ = barycentric quotient X_i/X_j for these {i,j}: {4, 20}, {19, 610}, {25, 154}, {34, 1394}, {64, 3}, {115, 1562}, {125, 122}, {158, 1895}, {225, 5930}, {235, 2883}, {253, 69}, {393, 1249}, {459, 2}, {1073, 394}, {1096, 204}, {1301, 110}, {1824, 3198}, {1895, 1097}, {2155, 48}, {2184, 63}, {2207, 3172}, {3183, 2060}, {5931, 332}

Notes

Let A'B'C' be the midheight triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₄₅₉. (Randy Hutson, June 7, 2019)