X₅₁₇

ISOGONAL CONJUGATE OF X₁₀₄

As the isogonal conjugate of a point on the circumcircle, X₅₁₇ lies on the line at infinity.

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

X₅₁₇ (ISOGONAL CONJUGATE OF X₁₀₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {4, 8}

• Related Centers: X₁₀₄ (ANTIPODE OF X₁₀₀), X₁ (INCENTER), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₈ (NAGEL POINT)

Trilinears: \(-1 + \cos B + \cos C : -1 + \cos C + \cos A : -1 + \cos A + \cos B\)

Barycentrics: \((\sin A)(-1 + \cos B + \cos C) : (\sin B)(-1 + \cos C + \cos A) : (\sin C)(-1 + \cos A + \cos B)\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \(b^{3} + c^{3} - (a^{2} + bc)(b + c) + 2abc : :\)

• Barycentrics: \((a - c) \cos B + (a - b) \cos C : :\)

▸ Linear Combinations1 combination

• X₅₁₇ = X₁ - X₃

▸ Incident Central Lines117 lines

{1, 3}, {2, 392}, {4, 8}, {5, 10}, {6, 998}, {7, 1000}, {9, 374}, {11, 1737}, {19, 219}, {20, 145}, {21, 1389}, {30, 511}, {33, 1905}, {34, 1753}, {37, 573}, {42, 1064}, {44, 1168}, {52, 1858}, {59, 1870}, {63, 956}, {71, 1243}, {74, 1290}, {78, 945}, {88, 1318}, {98, 2703}, {99, 2699}, {100, 953}, {101, 910}, {102, 1807}, {103, 1308}, {104, 901}, {105, 2742}, {106, 2743}, {107, 2744}, {108, 2745}, {109, 1455}, {110, 1325}, {111, 2746}, {112, 2747}, {119, 908}, {140, 1125}, {151, 1535}, {169, 220}, {182, 1386}, {201, 2599}, {210, 381}, {218, 2082}, {221, 3157}, {226, 495}, {238, 1052}, {244, 1149}, {347, 1439}, {376, 2094}, {389, 950}, {390, 3488}, {399, 2948}, {496, 1210}, {500, 2650}, {549, 551}, {550, 1483}, {572, 1100}, {579, 1108}, {580, 595}, {582, 602}, {601, 1468}, {672, 2170}, {840, 1292}, {906, 1951}, {936, 1706}, {938, 1058}, {958, 3560}, {990, 1350}, {997, 1376}, {1006, 1621}, {1042, 1066}, {1046, 2943}, {1051, 2944}, {1068, 1426}, {1113, 2103}, {1114, 2102}, {1124, 2362}, {1148, 1895}, {1293, 2718}, {1294, 2719}, {1295, 2720}, {1296, 2721}, {1297, 2722}, {1817, 3025}, {1352, 3416}, {1361, 1785}, {1362, 3328}, {1364, 3319}, {1391, 1443}, {1411, 2361}, {1437, 3193}, {1451, 1497}, {1457, 1465}, {1478, 1836}, {1479, 1837}, {1490, 2136}, {1656, 1698}, {1702, 3311}, {1703, 3312}, {1788, 3086}, {1830, 1877}, {1838, 1888}, {2171, 2269}, {2182, 2323}, {2270, 2324}, {2329, 3496}, {3022, 3322}, {3061, 3501}, {3085, 3485}, {3125, 3290}, {3190, 3198}, {3197, 3211}, {3474, 3476}

▸ Functional Relations16 relations

• X₅₁₇ = isogonal conjugate of X₁₀₄

• X₅₁₇ = isotomic conjugate of X₁₈₈₁₆

• X₅₁₇ = anticomplementary conjugate of X₁₅₃

• X₅₁₇ = complementary conjugate of X₁₁₉

• X₅₁₇ = X₂-Ceva conjugate of X₂₃₉₈₀

• X₅₁₇ = X₄-Ceva conjugate of X₁₁₉

• X₅₁₇ = crosspoint of X_i and X_j for these (i,j): (1, 80), (7, 88)

• X₅₁₇ = crosssum of X_i and X_j for these (i,j): (1, 36), (3, 912), (44, 55), (56, 1455)

• X₅₁₇ = crossdifference of every pair of points on line X₆X₆₅₀

• X₅₁₇ = orthopoint of X₅₁₃

• X₅₁₇ = trilinear pole of line X₁₇₆₉X₃₃₁₀

• X₅₁₇ = Thomson-isogonal conjugate of X₁₀₀

• X₅₁₇ = Lucas-isogonal conjugate of X₁₀₀

• X₅₁₇ = Cundy-Parry Phi transform of X₅₆

• X₅₁₇ = Cundy-Parry Psi transform of X₈

• X₅₁₇ = X₃₀-of-excentral-triangle