X₃₅₇₀

INTERSECTION OF LINES P(6)U(8) AND P(8)U(6)

The points given by trilinears P(6) =\(b : c : a\) and U(6) =\(c : a : b\) are indexed at Bicentric Pairs (accessible using the Tables button at the top of ETC), and likewise for their isogonal conjugates P(8) =\(1/b : 1/c : 1/a\) and U(8) =\(1/c : 1/a : 1/b\). The bicentric pair of lines P(6)U(8) and P(8)U(6) concur in X₃₅₇₀. Contributed by Peter Moses, July 7, 2009..

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

X₃₅₇₀ (INTERSECTION OF LINES P(6)U(8) AND P(8)U(6)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {100, 190}

• Related Centers: X₃₅₇₂ (INTERSECTION OF LINES P), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₁₉₀ (YFF PARABOLIC POINT)

Trilinears: \(bc(a^{2} - bc)/(b - c) : ca(b^{2} - ca)/(c - a) : ab(c^{2} - ab)/(a - b)\)

Barycentrics: \((a^{2} - bc)/(b - c) : (b^{2} - ca)/(c - a) : (c^{2} - ab)/(a - b)\)

Search Value (6, 9, 13): \(-9.047652695934\)

▸ Incident Central Lines8 lines

{2, 6}, {100, 190}, {101, 668}, {106, 3227}, {662, 799}, {666, 1026}, {1016, 1023}, {1914, 3253}

▸ Functional Relations6 relations

• X₃₅₇₀ = isogonal conjugate of X₃₅₇₂

• X₃₅₇₀ = isotomic conjugate of X₄₄₄₄

• X₃₅₇₀ = X₆₆₆-Ceva conjugate of X₁₉₀

• X₃₅₇₀ = cevapoint of X₆₅₉ and X₂₂₃₈

• X₃₅₇₀ = crossdifference of every pair of points on the line X₅₁₂X₁₀₁₅

• X₃₅₇₀ = trilinear pole of line X₂₃₈X₂₃₉