X₃₀₅₁

INTERSECTION X₂X_6∩X₃₂X_184

X₃₀₅₁ = X₂₅₃₁-cross conjugate of X₁₅₇₆

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

X₃₀₅₁ (INTERSECTION X₂X_6∩X₃₂X_184) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {32, 184}

• Related Centers: X₃₀₈ (ISOTOMIC CONJUGATE OF X₃₉), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₃₂ (3rd POWER POINT), X₁₈₄ (INVERSE OF X)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{3}(b^{2} + c^{2})\)

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

Search Value (6, 9, 13): \(0.510063673737\)

▸ Incident Central Lines15 lines

{2, 6}, {25, 263}, {31, 1911}, {32, 184}, {42, 1197}, {51, 1196}, {99, 703}, {110, 251}, {213, 2308}, {321, 2235}, {511, 1194}, {669, 881}, {1078, 1207}, {1180, 2979}, {1627, 1691}

▸ Functional Relations8 relations

• X₃₀₅₁ = isogonal conjugate of X₃₀₈

• X₃₀₅₁ = X_i-Ceva conjugate of X_j for these (i,j): (6, 39), (110, 669), (694, 237), (1576, 2531)

• X₃₀₅₁ = X₂₅₃₁-cross conjugate of X₁₅₇₆

• X₃₀₅₁ = crosspoint of X_i and X_j for these (i,j): (6, 32), (39, 1843)

• X₃₀₅₁ = crosssum of X_i and X_j for these (i,j): (2, 76), (83, 1799)

• X₃₀₅₁ = crossdifference of every pair of points on line X₃₁₆X₅₁₂ (the anticomplement of the Lemoine axis)

• X₃₀₅₁ = Danneels point of X₆

• X₃₀₅₁ = X₉₂-isoconjugate of X₁₇₉₉

Notes

Let L be the isogonal conjugate of the isotomic conjugate of the Brocard axis (i.e., line X₃₂X₁₈₄). Let M be the isotomic conjugate of the isogonal conjugate of the Brocard axis (i.e., line X₂X₆₉). Then X₃₀₅₁ = L∩M. (Randy Hutson, March 21, 2019)