X₁₅₃₂

ORTHOJOIN OF X₃₇

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

X₁₅₃₂ (ORTHOJOIN OF X₃₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {11, 515}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₁ (FEUERBACH POINT), X₅₁₅ (ISOGONAL CONJUGATE OF X₁₀₂)

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

Search Value (6, 9, 13): \(-10.895041318831\)

▸ Incident Central Lines6 lines

{2, 3}, {11, 515}, {12, 946}, {40, 1329}, {119, 517}, {496, 944}

▸ Functional Relations1 relation

• X₁₅₃₂ = reflection of X_i in X_j for these (i,j): (1519, 1538), (1537, 1519)

Notes

As a point on the Euler line, X₁₅₃₂ has Shinagawa coefficients \((\$a\$S^{2}, \$a\$S^{2} + 2abc(E + F) + 2\$a(S_{A})^{2}\$)\).