X₅₁₄

ISOGONAL CONJUGATE OF X₁₀₁

X₅₁₄ is the isogonal conjugate of X₁₀₁.

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

X₅₁₄ (ISOGONAL CONJUGATE OF X₁₀₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {1, 663}

• Related Centers: X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₁ (INCENTER), X₆₆₃ (CROSSDIFFERENCE OF X₂ AND X₇)

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

Barycentrics: \(b - c : :\)

▸ Alternate Coordinates3 forms

• Trilinears: \(|AP(1)| - |AU(1)| : :\)

• Trilinears: directed distance from A to line X₁X₆ : :

• Barycentrics: \(\cot B/2 - \cot C/2 : :\)

▸ Incident Central Lines49 lines

{1, 663}, {2, 1022}, {10, 764}, {11, 3328}, {30, 511}, {57, 2401}, {74, 2688}, {85, 2140}, {98, 2700}, {99, 2702}, {100, 1308}, {101, 664}, {102, 2723}, {103, 2724}, {104, 2717}, {105, 2725}, {106, 2726}, {107, 2727}, {108, 2728}, {109, 929}, {110, 2690}, {111, 2729}, {116, 1146}, {189, 2399}, {190, 1016}, {239, 649}, {241, 650}, {242, 1459}, {330, 3249}, {651, 655}, {653, 1461}, {659, 667}, {661, 693}, {1024, 2402}, {1111, 2170}, {1292, 2736}, {1293, 2737}, {1294, 2738}, {1295, 2739}, {1296, 2740}, {1297, 2741}, {1317, 3322}, {1358, 3323}, {1729, 3188}, {1734, 2254}, {1768, 2958}, {1921, 3261}, {2487, 2516}, {4521, 4885}

▸ Functional Relations31 relations

• X₅₁₄ = isogonal conjugate of X₁₀₁

• X₅₁₄ = isotomic conjugate of X₁₉₀

• X₅₁₄ = anticomplementary conjugate of X₁₅₀

• X₅₁₄ = complementary conjugate of X₁₁₆

• X₅₁₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 1086), (4, 116), (7, 11), (75, 244), (100, 142), (190, 2)

• X₅₁₄ = X_i-cross conjugate of X_j for these (i,j): (11, 7), (244, 75)

• X₅₁₄ = crosspoint of X₂ and X₁₉₀

• X₅₁₄ = crosssum of X_i and X_j for these (i,j): (6, 649), (37, 650), (41, 663), (48, 652), (55, 657), (213, 667), (354, 513), (1459, 1473)

• X₅₁₄ = crossdifference of every pair of points on line X₆X₃₁

• X₅₁₄ = orthopoint of X₅₁₆

• X₅₁₄ = excentral-isogonal conjugate of X₃₉₁₅₆

• X₅₁₄ = X₅₁₃-Hirst inverse of X₈₁₂

• X₅₁₄ = trilinear pole of the line X₁₁X₂₄₄ (which is the Feuerbach tangent line)

• X₅₁₄ = pole wrt polar circle of line X₄X₉

• X₅₁₄ = X₄₈-isoconjugate (polar conjugate) of X₁₈₉₇

• X₅₁₄ = X₆-isoconjugate of X₁₀₀

• X₅₁₄ = intercept of Gergonne line and the line at infinity (trilinear polars of X₇ and X₂)

• X₅₁₄ = X₆₉₀ of Fuhrmann triangle

• X₅₁₄ = bicentric difference of PU(i) for these i: 10, 24, 47, 51

• X₅₁₄ = ideal point of PU(i) for these i: 10, 47, 51

• X₅₁₄ = trilinear product of PU(27)

• X₅₁₄ = trilinear pole of PU(i) for these i: 121, 123

• X₅₁₄ = perspector of circumconic centered at X₁₀₈₆ (hyperbola {A, B, C,X₂,X₇}})

• X₅₁₄ = center of circumconic that is locus of trilinear poles of lines passing through X₁₀₈₆

• X₅₁₄ = anticomplementary isotomic conjugate of X₅₁₄

• X₅₁₄ = Thomson-isogonal conjugate of X₁₀₃

• X₅₁₄ = Lucas-isogonal conjugate of X₁₀₃

• X₅₁₄ = barycentric square root of X₁₀₈₆

• X₅₁₄ = X₆₃-isoconjugate of X₈₇₅₀

• X₅₁₄ = X₉₂-isoconjugate of X₃₂₆₅₆

• X₅₁₄ = perspector of ABC and vertex-triangle of Gemini triangles 7 and 8

Notes

Let A₇B₇C₇ and A₈B₈C₈ be Gemini triangles 7 and 8, resp. Let L_A be the tangent at A to conic {A,B₇,C₇,B₈,C₈}}, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₅₁₄. (Randy Hutson, January 15, 2019)

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