X₅₁₈

ISOGONAL CONJUGATE OF X₁₀₅

As the isogonal conjugate of a point on the circumcircle, X₅₁₈ lies on the line at infinity.

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

X₅₁₈ (ISOGONAL CONJUGATE OF X₁₀₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {7, 8}

• Related Centers: X₁₀₅ (Λ(INCENTER, SYMMEDIAN POINT)), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₇ (GERGONNE POINT), X₈ (NAGEL POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = ab + ac - b^{2} - c^{2}\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

▸ Alternate Coordinates1 form

• Trilinears: \(as - S_{\omega } : bs - S_{\omega } : cs - S_{\omega }\)

▸ Incident Central Lines95 lines

{1, 6}, {2, 210}, {3, 3433}, {7, 8}, {10, 141}, {11, 908}, {20, 3189}, {21, 2346}, {30, 511}, {36, 42}, {38, 42}, {40, 1071}, {43, 982}, {55, 63}, {56, 78}, {57, 200}, {59, 765}, {74, 2691}, {80, 3254}, {81, 1390}, {92, 1859}, {98, 2704}, {99, 2711}, {100, 840}, {101, 2725}, {102, 2730}, {103, 2736}, {104, 2742}, {105, 1280}, {106, 2748}, {107, 2749}, {108, 2750}, {109, 2751}, {110, 2752}, {111, 2753}, {112, 2754}, {144, 145}, {165, 3158}, {182, 1385}, {209, 306}, {226, 2886}, {239, 335}, {241, 1458}, {243, 1897}, {244, 899}, {318, 1887}, {329, 497}, {355, 1352}, {474, 3338}, {551, 597}, {583, 1009}, {612, 940}, {643, 2651}, {651, 1456}, {668, 1921}, {677, 1814}, {869, 2274}, {872, 1193}, {896, 902}, {910, 1281}, {936, 3333}, {938, 2551}, {959, 1219}, {961, 1257}, {976, 1468}, {997, 999}, {1156, 1320}, {1210, 1329}, {1214, 3190}, {1222, 1431}, {1260, 1617}, {1331, 2361}, {1351, 1482}, {1353, 1483}, {1362, 3323}, {1478, 3419}, {1621, 3219}, {1706, 3339}, {1707, 3052}, {1824, 1889}, {1829, 1843}, {1836, 3434}, {1837, 3436}, {1861, 1876}, {1992, 3241}, {2076, 3099}, {2093, 2097}, {2102, 2104}, {2103, 2105}, {2136, 2951}, {2223, 3286}, {2238, 3290}, {2292, 2667}, {2330, 2646}, {2930, 2948}

▸ Functional Relations22 relations

• X₅₁₈ = isogonal conjugate of X₁₀₅

• X₅₁₈ = complementary conjugate of X₁₂₀

• X₅₁₈ = X_i-Ceva conjugate of X_j for these (i,j): (2, 6184), (4, 120), (335, 37)

• X₅₁₈ = crosspoint of X₁ and X₂₉₁

• X₅₁₈ = crosssum of X_i and X_j for these (i,j): (1, 238), (56, 1456)

• X₅₁₈ = crossdifference of every pair of points on line X₆X₅₁₃

• X₅₁₈ = X_i-Hirst inverse of X_j for these (i,j): (1, 9), (6, 1083)

• X₅₁₈ = X_i-line conjugate of X_j for these (i,j): (1, 6), (30, 513)

• X₅₁₈ = isotomic conjugate of X₂₄₈₁

• X₅₁₈ = X₂₇₈₁ of Fuhrmann triangle

• X₅₁₈ = ideal point of PU(28)

• X₅₁₈ = crossdifference of PU(i) for i in (46, 54)

• X₅₁₈ = X₆-isoconjugate of X₆₇₃

• X₅₁₈ = trilinear pole of line X₆₆₅X₁₆₄₂

• X₅₁₈ = perspector of conic {A, B, C,X₂,X₁₀₀,PU(112)}

• X₅₁₈ = trilinear square root of X₄₇₁₂

• X₅₁₈ = Cundy-Parry Psi transform of X₁₄₂₆₈

• X₅₁₈ = Thomson-isogonal conjugate of X₁₂₉₂

• X₅₁₈ = X₆₃-isoconjugate of X₈₇₅₁

• X₅₁₈ = X₉₂-isoconjugate of X₃₂₆₅₈

• X₅₁₈ = Lucas-isogonal conjugate of X₁₂₉₂

• X₅₁₈ = polar conjugate of isotomic conjugate of X₂₅₀₈₃

Notes

Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b and I_c cyclically; then X₅₁₈ = X₅₂₈ of I_aI_bI_c. (Randy Hutson, December 26, 2015)