X₅₁₆

ISOGONAL CONJUGATE OF X₁₀₃

X₅₁₆ is the perspector of triangle ABC and the tangential triangles of the conic that passes through the points A, B, C, X₂, and X₇.

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

X₅₁₆ (ISOGONAL CONJUGATE OF X₁₀₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 7}, {4, 9}

• Related Centers: X₁₀₃ (ANTIPODE OF X₁₀₁), X₁ (INCENTER), X₇ (GERGONNE POINT), X₄ (ORTHOCENTER), X₉ (MITTENPUNKT)

Trilinears: \(bc[(a - b) \cot C + (a - c) \cot B] : :\)

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

▸ Alternate Coordinates2 forms

• Trilinears: \(a^{2} - b^{2} \cos C - c^{2} \cos B : :\)

• Trilinears: \(a^{2} - b^{2} \cos C - c^{2} \cos B : :\)

▸ Linear Combinations1 combination

• X₅₁₆ = X₁ - X₇

▸ Incident Central Lines91 lines

{1, 7}, {2, 165}, {3, 142}, {4, 9}, {8, 144}, {11, 1155}, {27, 2328}, {30, 511}, {31, 1754}, {35, 411}, {46, 1210}, {55, 226}, {57, 497}, {63, 1709}, {65, 950}, {72, 3059}, {74, 2690}, {79, 2346}, {80, 655}, {98, 2702}, {99, 2700}, {100, 908}, {101, 2724}, {102, 929}, {103, 927}, {104, 1308}, {105, 2736}, {106, 2737}, {107, 2738}, {108, 2739}, {109, 1936}, {110, 2688}, {111, 2740}, {112, 2741}, {118, 910}, {146, 2948}, {149, 1768}, {152, 1282}, {200, 329}, {214, 1537}, {238, 673}, {255, 1777}, {354, 553}, {355, 382}, {376, 551}, {388, 1697}, {412, 1838}, {484, 1737}, {550, 1385}, {580, 3073}, {902, 3011}, {938, 3339}, {944, 3243}, {972, 1543}, {993, 1012}, {1058, 3333}, {1076, 1771}, {1086, 1279}, {1158, 3358}, {1284, 2223}, {1292, 2725}, {1293, 2726}, {1294, 2727}, {1295, 2728}, {1296, 2729}, {1317, 3328}, {1376, 3452}, {1389, 3255}, {1482, 1657}, {1490, 3174}, {1519, 2077}, {1538, 3035}, {1571, 2548}, {1572, 2549}, {1587, 1702}, {1588, 1703}, {1633, 3220}, {1698, 3091}, {1700, 2546}, {1701, 2547}, {1704, 2542}, {1705, 2543}, {1736, 2310}, {1829, 1885}, {1852, 1888}, {2017, 2544}, {2018, 2545}, {2321, 3416}, {2947, 3190}, {3021, 3323}, {3340, 3586}

▸ Functional Relations18 relations

• X₅₁₆ = isogonal conjugate of X₁₀₃

• X₅₁₆ = isotomic conjugate of X₁₈₀₂₅

• X₅₁₆ = anticomplementary conjugate of X₁₅₂

• X₅₁₆ = complementary conjugate of X₁₁₈

• X₅₁₆ = X₂-Ceva conjugate of X₂₃₉₇₂

• X₅₁₆ = X₄-Ceva conjugate of X₁₁₈

• X₅₁₆ = crosssum of X_i and X_j for these (i,j): (3, 916), (55, 672)

• X₅₁₆ = crossdifference of every pair of points on line X₆X₆₅₇

• X₅₁₆ = orthopoint of X₅₁₄

• X₅₁₆ = intercept of the Soddy line and the line at infinity (trilinear polars of X₇ and X₂)

• X₅₁₆ = X₅₄₂-of-Fuhrmann-triangle

• X₅₁₆ = exsimilicenter of Bevan and anticomplementary circles

• X₅₁₆ = harmonic center of Bevan and anticomplementary circles

• X₅₁₆ = Thomson-isogonal conjugate of X₁₀₁

• X₅₁₆ = Lucas-isogonal conjugate of X₁₀₁

• X₅₁₆ = Cundy-Parry Phi transform of X₁₄₃₇₇

• X₅₁₆ = Cundy-Parry Psi transform of X₃₇₃₀

• X₅₁₆ = X₉₂-isoconjugate of X₃₂₆₅₇

Notes

X₅₁₆ is one of two centers of inverse similitude (X₅₁₄ is the other) of the 1st and 2nd Dao equilateral triangles. (Randy Hutson, July 11, 2019)

As the isogonal conjugate of a point on the circumcircle, X₅₁₆ lies on the line at infinity.