X₁₃₅

INTERSECTION OF LINE X₄X_131 AND X₂₅X_114

X₁₃₅ = X₁₀₈-of-orthic-triangle if ABC is acute

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

X₁₃₅ (INTERSECTION OF LINE X₄X_131 AND X₂₅X_114) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 131}, {25, 114}

• Related Centers: X₄ (ORTHOCENTER), X₁₃₁ (INTERSECTION OF LINES X), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₁₁₄ (KIEPERT ANTIPODE)

Trilinears: \((\sec A)[(\sin 2B)/(\sec 2C - \sec 2A) + (\sin 2C)/(\sec 2A - \sec 2B)] : :\)

Barycentrics: \((\tan A)[(\sin 2B)/(\sec 2C - \sec 2A) + (\sin 2C)/(\sec 2A - \sec 2B)] : :\)

Search Value (6, 9, 13): \(0.389229429446\)

X₁₃₅ lies on the nine-point circle

▸ Incident Central Lines4 lines

{4, 131}, {25, 114}, {52, 113}, {119, 431}

▸ Functional Relations7 relations

• X₁₃₅ = X₁₀₈-of-orthic-triangle if ABC is acute

• X₁₃₅ = center of the rectangular hyperbola that passes through A, B, C, and X₂₄

• X₁₃₅ = anticomplement of X₃₄₈₄₃

• X₁₃₅ = inverse-in-polar-circle of X₉₂₅

• X₁₃₅ = crosssum of circumcircle intercepts of line X₃X₆₈

• X₁₃₅ = orthopole of line X₃X₆₈

• X₁₃₅ = Kirikami-six-circles-image of X₂₄