X₅₁₅

ISOGONAL CONJUGATE OF X₁₀₂

X₅₁₅ is the perspector of triangle ABC and the tangential triangles of the conic that passes through the points A, B, C, X₂, and X₈.

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

X₅₁₅ (ISOGONAL CONJUGATE OF X₁₀₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {3, 10}

• Related Centers: X₁₀₂ (Λ(INCENTER, ORTHOCENTER)), X₁ (INCENTER), X₄ (ORTHOCENTER), X₃ (CIRCUMCENTER), X₁₀ (SPIEKER CENTER)

Trilinears: \((b + c)\sec A - b \sec B - c \sec C :\)

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

▸ Alternate Coordinates1 form

• Trilinears: \(r - 2 R \cos B \cos C : :\)

▸ Linear Combinations1 combination

• X₅₁₅ = X₁ - X₄

▸ Incident Central Lines73 lines

{1, 4}, {3, 10}, {5, 1125}, {8, 20}, {9, 3427}, {11, 1319}, {12, 2646}, {29, 947}, {30, 511}, {36, 80}, {48, 1826}, {55, 1012}, {56, 1210}, {58, 3072}, {65, 1071}, {71, 1765}, {74, 2689}, {78, 3436}, {79, 1389}, {98, 2701}, {99, 2708}, {100, 2077}, {101, 2723}, {102, 1309}, {103, 929}, {105, 2730}, {106, 2731}, {108, 2733}, {109, 2734}, {110, 2695}, {111, 2735}, {117, 1455}, {119, 214}, {145, 962}, {153, 908}, {165, 376}, {200, 3421}, {281, 610}, {284, 1065}, {381, 551}, {382, 1482}, {411, 2975}, {484, 1768}, {595, 3073}, {602, 1724}, {603, 1771}, {631, 1698}, {910, 1146}, {936, 2551}, {938, 3333}, {956, 3419}, {997, 3452}, {1000, 3062}, {1006, 1746}, {1292, 2751}, {1293, 2757}, {1294, 2762}, {1295, 2765}, {1296, 2768}, {1317, 1537}, {1323, 1565}, {1350, 3416}, {1387, 1538}, {1420, 8086}, {1498, 3173}, {1766, 2321}, {1829, 3575}, {1836, 2099}, {1839, 1963}, {1885, 1902}, {2093, 2096}, {2183, 2250}, {3241, 3543}

▸ Functional Relations14 relations

• X₅₁₅ = isogonal conjugate of X₁₀₂

• X₅₁₅ = isotomic conjugate of X₃₄₃₉₃

• X₅₁₅ = anticomplementary conjugate of X₁₅₁

• X₅₁₅ = complementary conjugate of X₁₁₇

• X₅₁₅ = X₂-Ceva conjugate of X₂₃₉₈₆

• X₅₁₅ = X₄-Ceva conjugate of X₁₁₇

• X₅₁₅ = crossdifference of every pair of points on line X₆X₆₅₂

• X₅₁₅ = orthopoint of X₅₂₂

• X₅₁₅ = intersection of trilinear polars of X₂ and X₈

• X₅₁₅ = Thomson-isogonal conjugate of X₁₀₉

• X₅₁₅ = Lucas-isogonal conjugate of X₁₀₉

• X₅₁₅ = Cundy-Parry Phi transform of X₁₀₅₇₀

• X₅₁₅ = Cundy-Parry Psi transform of X₁₀₅₇₁

• X₅₁₅ = X₂-isoconjugate of X₃₂₆₇₇

Notes

As the isogonal conjugate of a point on the circumcircle, X₅₁₅ lies on the line at infinity.