X₅₁₃

ISOGONAL CONJUGATE OF X₁₀₀

As the isogonal conjugate of a point on the circumcircle, X₅₁₃ lies on the line at infinity.

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

X₅₁₃ (ISOGONAL CONJUGATE OF X₁₀₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {36, 238}, {30, 511}

• Related Centers: X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₂₃₈ (-LINE CONJUGATE OF X₃₇), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈)

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

Barycentrics: \(ab - ac : bc - ba : ca - cb\)

▸ Alternate Coordinates2 forms

• Trilinears: \(r - R \cos A : :\)

• Trilinears: \(d(a,b,c) : :\), where d(a,b,c) = directed distance from A to the Nagel line

▸ Incident Central Lines59 lines

{1, 764}, {3, 3657}, {6, 1024}, {7, 885}, {9, 3126}, {11, 3025}, {30, 511}, {36, 238}, {37, 876}, {44, 649}, {59, 651}, {74, 2687}, {98, 2699}, {99, 2703}, {100, 765}, {101, 1308}, {102, 2716}, {103, 2717}, {104, 953}, {105, 840}, {106, 2718}, {107, 2719}, {108, 2720}, {109, 2222}, {110, 1290}, {111, 2721}, {112, 2711}, {190, 660}, {269, 2424}, {320, 350}, {484, 1734}, {663, 855}, {668, 889}, {676, 2488}, {884, 3423}, {927, 1275}, {957, 2401}, {1037, 1486}, {1052, 1054}, {1086, 3271}, {1292, 2742}, {1293, 2743}, {1294, 2744}, {1295, 2745}, {1296, 2746}, {1297, 2747}, {1361, 3319}, {1362, 3322}, {1364, 3326}, {1430, 2201}, {1835, 1874}, {1960, 3246}, {2473, 2487}, {2490, 2505}, {2500, 2532}, {2517, 2533}, {2529, 3239}, {3022, 3328}, {3123, 3248}

▸ Functional Relations37 relations

• X₅₁₃ = isogonal conjugate of X₁₀₀

• X₅₁₃ = isotomic conjugate of X₆₆₈

• X₅₁₃ = anticomplementary conjugate of X₁₄₉

• X₅₁₃ = complementary conjugate of X₁₁

• X₅₁₃ = X_i-Ceva conjugate of X_j for these (i,j): (1, 244), (4, 11), (100, 1), (101, 354), (108, 56), (109, 65), (190, 37), (38275, 23505)

• X₅₁₃ = X₂₄₄-cross conjugate of X₁

• X₅₁₃ = crosspoint of X_i and X_j for these (i,j): (1, 100), (4, 108), (58, 109), (86, 190)

• X₅₁₃ = crosssum of X_i and X_j for these (i,j): (1, 513), (3, 521), (6, 667), (10, 522), (42, 649), (55, 650), (142, 514), (442, 523), (692, 906), (900, 1145)

• X₅₁₃ = crossdifference of every pair of points on line X₁X₆

• X₅₁₃ = orthopoint of X₅₁₇

• X₅₁₃ = X_i-line conjugate of X_j for these (i,j): (30, 518), (36, 238)

• X₅₁₃ = X₅₂₆-of-Fuhrmann triangle

• X₅₁₃ = barycentric product of PU(27)

• X₅₁₃ = trilinear pole of PU(i) for these i: 27, 34

• X₅₁₃ = center of circumconic that is locus of trilinear poles of lines passing through X₁₀₁₅

• X₅₁₃ = X₂-Ceva conjugate of X₁₀₁₅

• X₅₁₃ = crossdifference of PU(28)

• X₅₁₃ = ideal point of PU(i), for these i: 3, 6, 24, 31, 33, 41, 46, 50, 52, 53, 54, 55, 56, 57, 58, 74, 76, 78, 96, 111, 124

• X₅₁₃ = bicentric difference of PU(i) for these i: 6, 31, 33, 41, 46, 50, 52, 53, 54, 55, 56, 96, 111

• X₅₁₃ = trilinear product of PU(34)

• X₅₁₃ = trilinear square root of X₂₄₄

• X₅₁₃ = perspector of circumconic centered at X₁₀₁₅ (hyperbola {A, B, C,X₁,X₂}

• X₅₁₃ = intercept of antiorthic axis and the line at infinity (trilinear polars of X₁ and X₂)

• X₅₁₃ = excentral isogonal conjugate of X₁₇₆₈

• X₅₁₃ = intouch isogonal conjugate of X₁₁

• X₅₁₃ = orthic isogonal conjugate of X₁₁

• X₅₁₃ = X₆-isoconjugate of X₁₉₀

• X₅₁₃ = X₁-vertex conjugate of X₃₆

• X₅₁₃ = barycentric cube root of X₈₀₂₇

• X₅₁₃ = trilinear pole of line X₂₄₄X₆₆₅

• X₅₁₃ = Thomson-isogonal conjugate of X₁₀₄

• X₅₁₃ = Lucas-isogonal conjugate of X₁₀₄

• X₅₁₃ = Cundy-Parry Psi transform of X₁₄₂₆₆

• X₅₁₃ = Cundy-Parry Phi transform of X₃₉₁₇₃

• X₅₁₃ = polar conjugate of X₆₃₃₅

• X₅₁₃ = pole wrt polar circle of trilinear polar of X₆₃₃₅ (line X₄X₈)

• X₅₁₃ = perspector of ABC and side-triangle of Gemini triangles 5 and 6