X₅₁₉

ISOGONAL CONJUGATE OF X₁₀₆

As the isogonal conjugate of a point on the circumcircle, X₅₁₉ lies on the line at infinity.

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

X₅₁₉ (ISOGONAL CONJUGATE OF X₁₀₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {36, 100}

• Related Centers: X₁₀₆ (Λ(INCENTER, CENTROID)), X₁ (INCENTER), X₂ (CENTROID), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT)

Trilinears: \((2a - b - c)/a : (2b - c - a)/b : (2c - a - b)/c\)

Barycentrics: \(2a - b - c : 2b - c - a : 2c - a - b\)

▸ Alternate Coordinates2 forms

• Trilinears: \(1 - 2 \csc A/2 \sin B/2 \sin C/2 : :\)

• Trilinears: \(3 r - 2 R \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₅₁₉ = X₁ - X₂

▸ Incident Central Lines89 lines

{1, 2}, {6, 996}, {9, 1000}, {30, 511}, {35, 2975}, {36, 100}, {37, 1573}, {40, 376}, {44, 2325}, {55, 956}, {57, 3476}, {58, 1043}, {63, 1727}, {65, 553}, {72, 950}, {74, 2692}, {80, 908}, {98, 2705}, {99, 2712}, {101, 2726}, {102, 2731}, {103, 2737}, {104, 2077}, {105, 2748}, {106, 1120}, {107, 2755}, {108, 2756}, {109, 2757}, {110, 2758}, {111, 2759}, {112, 2760}, {121, 3544}, {188, 1128}, {210, 392}, {214, 1145}, {226, 2099}, {238, 765}, {244, 1739}, {291, 3227}, {320, 668}, {346, 1743}, {350, 668}, {355, 381}, {388, 3340}, {405, 3303}, {428, 1829}, {447, 648}, {474, 3304}, {484, 3218}, {495, 2886}, {496, 1329}, {497, 3421}, {549, 1385}, {573, 3169}, {594, 1100}, {595, 2985}, {597, 1386}, {599, 3242}, {664, 1323}, {666, 1121}, {672, 1018}, {751, 984}, {958, 3295}, {962, 3543}, {966, 3247}, {999, 1376}, {1015, 1575}, {1056, 2550}, {1058, 2551}, {1107, 1500}, {1126, 1220}, {1150, 2177}, {1377, 3297}, {1378, 3298}, {1387, 3036}, {1420, 1788}, {1449, 2345}, {1478, 3434}, {1479, 3436}, {1697, 1776}, {1706, 3333}, {1785, 1897}, {1834, 3454}, {1837, 2098}, {1861, 1870}, {1862, 1878}, {2093, 2094}, {2654, 3191}, {3158, 3524}

▸ Functional Relations25 relations

• X₅₁₉ = isogonal conjugate of X₁₀₆

• X₅₁₉ = isotomic conjugate of X₉₀₃

• X₅₁₉ = complementary conjugate of X₁₂₁

• X₅₁₉ = anticomplementary conjugate of X₂₁₂₉₀

• X₅₁₉ = X_i-Ceva conjugate of X_j for these (i,j): (2, 4370). (4, 121), (80, 10)

• X₅₁₉ = crosssum of X_i and X_j for these (i,j): (6, 902), (56, 1457)

• X₅₁₉ = crossdifference of every pair of points on line X₆X₆₄₉

• X₅₁₉ = X_i-Hirst inverse of X_j for these (i,j): (513, 537), (514, 545)

• X₅₁₉ = X₂₇₇₇-of-Fuhrmann-triangle

• X₅₁₉ = bicentric sum of PU(24)

• X₅₁₉ = intercept of Nagel line and the line at infinity

• X₅₁₉ = trilinear pole of line X₉₀₀X₁₆₃₅

• X₅₁₉ = X₆-isoconjugate of X₈₈

• X₅₁₉ = trilinear square root of X₄₇₃₈

• X₅₁₉ = barycentric cube root of X₈₀₂₈

• X₅₁₉ = Cundy-Parry Psi transform of X₁₄₂₆₁

• X₅₁₉ = Thomson-isogonal conjugate of X₁₂₉₃

• X₅₁₉ = Lucas-isogonal conjugate of X₁₂₉₃

• X₅₁₉ = polar conjugate of X₆₃₃₆

• X₅₁₉ = pole wrt polar circle of trilinear polar of X₆₃₃₆ (line X₄X₂₄₅₇)

• X₅₁₉ = X₁₉-isoconjugate of X₁₇₉₇

• X₅₁₉ = X₆₃-isoconjugate of X₈₇₅₂

• X₅₁₉ = X₉₂-isoconjugate of X₃₂₆₅₉

• X₅₁₉ = barycentric square root of X₄₃₇₀

• X₅₁₉ = homothetic center of Gemini triangle 28 and cross-triangle of Gemini triangles 20 and 28

Notes

Let I_aI_bI_c be the incentral triangle. Let A″ be the reflection of A in I_a, and define B″ and C″ cyclically. Let A* be the cevapoint of B″ and C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₅₁₉. (Randy Hutson, December 26, 2015)

Let A'B'C' be triangle T(-2,1). Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₁₉. The lines A'A″, B'B″, C'C″ concur in X₈. (Randy Hutson, December 26, 2015)