X₁₃₇

X₁₁₀-OF-ORTHIC-TRIANGLE

X₁₃₇ = X₁₁₀-of-orthic triangle

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

X₁₃₇ (X₁₁₀-OF-ORTHIC-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Concurrence Lines: H_a–X₁₃₇, H_b–X₁₃₇, H_c–X₁₃₇ concurring at X₁₃₇

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sec A)(\sin 2B + \sin 2C)[(\sin 2B - \sin 2C)^{2}] u(A,B,C)\), \(u(A,B,C) = [\sin^{2}(2A) - \sin^{2}(2B) - \sin^{2}(2C) - \sin 2B \sin 2C]\)

Barycentrics: \(af(A,B,C) : bf(B,C,A) : cf(C,A,B)\)

Search Value (6, 9, 13): \(-3.125748576905\)

X₁₃₇ lies on the nine-point circle

X₁₃₇ lies on the cevian circle of X₅.

▸ Incident Central Lines5 lines

{5, 128}, {51, 129}, {53, 138}, {113, 546}, {132, 428}

▸ Functional Relations12 relations

• X₁₃₇ = X₁₁₀-of-orthic triangle

• X₁₃₇ = center of the rectangular hyperbola that passes through A, B, C, X₅, and X₅₃

• X₁₃₇ = reflection of X₁₂₈ in X₅

• X₁₃₇ = complement of X₉₃₀

• X₁₃₇ = X₄-Ceva conjugate of X₁₅₁₀

• X₁₃₇ = crosssum of X₂₅₂ and X₉₃₀

• X₁₃₇ = trilinear pole wrt orthic triangle of line X₅X₅₃

• X₁₃₇ = inverse-in-polar-circle of X₉₃₃

• X₁₃₇ = excentral-to-ABC functional image of X₁₁₀

• X₁₃₇ = crosssum of circumcircle intercepts of line X₃X₅₄

• X₁₃₇ = orthopole of line X₃X₅₄

• X₁₃₇ = Kirikami-six-circles image of X₅