X₁₅₁₃

ORTHOJOIN OF X₂

X₁₅₁₃ = orthopole of Lemoine axis

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

X₁₅₁₃ (ORTHOJOIN OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {98, 230}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₉₈ (TARRY POINT), X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2})[3a^{4} + (b^{2} - c^{2})^{2}]\)

Search Value (6, 9, 13): \(-9.742191610428\)

▸ Incident Central Lines7 lines

{2, 3}, {98, 230}, {114, 325}, {132, 232}, {147, 385}, {183, 1352}, {1181, 1184}

▸ Functional Relations5 relations

• X₁₅₁₃ = reflection of X_i in X_j for these (i,j): (98, 230), (325, 114)

• X₁₅₁₃ = circumcircle-inverse of X₃₇₉₃₀

• X₁₅₁₃ = {X₂,X₃}-harmonic conjugate of X₃₇₄₅₀

• X₁₅₁₃ = orthopole of Lemoine axis

• X₁₅₁₃ = X₁₁₄-of-1st-anti-Brocard-triangle

Notes

As a point on the Euler line, X₁₅₁₃ has Shinagawa coefficients \((S^{2}, -(E + F)^{2})\).