X₃₁₂₄

DANNEELS PERSPECTOR FOR X₇₆

X₃₁₂₄ = crosssum of circumcircle intercepts of line X₂X₃₂

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

X₃₁₂₄ (DANNEELS PERSPECTOR FOR X₇₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 110}, {115, 125}

• Related Centers: X₆ (SYMMEDIAN POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₁₂₅ (CENTER OF JERABEK HYPERBOLA)

Trilinears: \(a(b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(1.070726738365\)

X₃₁₂₄ lies on the Brocard inellipse.

▸ Incident Central Lines22 lines

{2, 694}, {6, 110}, {23, 1691}, {25, 1501}, {32, 3457}, {39, 373}, {42, 2054}, {51, 1196}, {115, 125}, {237, 2021}, {244, 661}, {351, 865}, {394, 2987}, {593, 2248}, {1193, 2653}, {1495, 1692}, {1506, 3118}, {1613, 3060}, {1994, 2056}, {2092, 3030}, {2501, 2970}, {2679, 3005}

▸ Functional Relations23 relations

• X₃₁₂₄ = isogonal conjugate of X₄₅₉₀

• X₃₁₂₄ = isotomic conjugate of X₃₄₅₃₇

• X₃₁₂₄ = complement of X₄₅₇₆

• X₃₁₂₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 3005), (6, 512), (26, 669), (76, 523), (83, 2514), (111, 351), (755, 2872), (2207, 2489), (2248, 649), (3125, 2643)

• X₃₁₂₄ = X_i-cross conjugate of X_j for these (i,j): (1084, 2971), (2679, 2086)

• X₃₁₂₄ = crosspoint of X_i and X_j for these (i,j): (6, 512), (25, 2501), (76, 523), (647, 2351), (661, 756), (2207, 2489), (3122, 3125)

• X₃₁₂₄ = crosssum of X_i and X_j for these (i,j): (2, 99), (6, 1634), (32, 110), (112, 459), (317, 648), (662, 757), (6189, 6190)

• X₃₁₂₄ = crosssum of circumcircle intercepts of line X₂X₃₂

• X₃₁₂₄ = X_i-line conjugate of X_j for these (i,j): 6,110 111,110

• X₃₁₂₄ = crossdifference of every pair of points on the line X₉₉X₁₁₀

• X₃₁₂₄ = perspector of circumconic centered at X₃₀₀₅

• X₃₁₂₄ = center of circumconic that is locus of trilinear poles of lines passing through X₃₀₀₅ (the hyperbola {{A, B, C,X₆,X₃₉}})

• X₃₁₂₄ = trilinear pole wrt symmedial triangle of Brocard axis

• X₃₁₂₄ = intersection of tangents to Steiner inellipse at X₁₁₅ and X₁₀₈₄

• X₃₁₂₄ = crosspoint wrt medial triangle of X₁₁₅ and X₁₀₈₄

• X₃₁₂₄ = bicentric difference of PU(105)

• X₃₁₂₄ = PU(105)-harmonic conjugate of X₃₅₁

• X₃₁₂₄ = pole of de Longchamps line wrt Kiepert hyperbola

• X₃₁₂₄ = perspector of unary cofactor triangles of 1st and 2nd Brocard triangles

• X₃₁₂₄ = crosspoint of X₅₆₃₉ and X₅₆₃₈

• X₃₁₂₄ = barycentric square of X₆₆₁

• X₃₁₂₄ = polar conjugate of isotomic conjugate of X₂₀₉₇₅

• X₃₁₂₄ = X₆₃-isoconjugate of X₁₈₀₂₀