X₁₃₆

INTERSECTION OF LINE X₄X_110 AND X₂₅X_132

X₁₃₆ =X₁₀₉-of-orthic triangle if ABC is acute

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

X₁₃₆ (INTERSECTION OF LINE X₄X_110 AND X₂₅X_132) 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–X₁₃₆, H_b–X₁₃₆, H_c–X₁₃₆ concurring at X₁₃₆

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

Barycentrics: \((b^{2} - c^{2})^{2} (a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2})/(a^{2} - b^{2} - c^{2}) : :\)

Search Value (6, 9, 13): \(4.219605641607\)

▸ Alternate Coordinates1 form

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

X₁₃₆ lies on the nine-point circle

▸ Incident Central Lines12 lines

{2, 925}, {4, 110}, {5, 131}, {25, 132}, {68, 254}, {114, 427}, {117, 407}, {118, 430}, {119, 429}, {125, 338}, {127, 868}, {133, 235}

▸ Functional Relations18 relations

• X₁₃₆ =X₁₀₉-of-orthic triangle if ABC is acute

• X₁₃₆ = center of the rectangular hyperbola that passes through A, B, C, and X₉₃

• X₁₃₆ = reflection of X₁₃₁ in X₅

• X₁₃₆ = complement of X₉₂₅

• X₁₃₆ = anticomplement of X₃₄₈₄₄

• X₁₃₆ = crosssum of circumcircle intercepts of line X₃X₄₉

• X₁₃₆ = complementary conjugate of X₉₂₄

• X₁₃₆ = X₂₅₄-Ceva conjugate of X₅₂₃

• X₁₃₆ = perspector of circumconic centered at X₂₅₀₁

• X₁₃₆ = center of circumconic that is locus of trilinear poles of lines passing through X₂₅₀₁ (hyperbola {A, B, C,X₄,X₉₃}})

• X₁₃₆ = X₂-Ceva conjugate of X₂₅₀₁

• X₁₃₆ = inverse-in-polar-circle of X₁₁₀

• X₁₃₆ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₃₅₆₃

• X₁₃₆ = orthopole of line X₃X₄₉

• X₁₃₆ = Kirikami-six-circles image of X₉₃

• X₁₃₆ = Dou-circles-radical-circle-inverse of X₃₆₄₇₂

• X₁₃₆ = Moses-radical-circle-inverse of X₃₈₉₇₀

• X₁₃₆ = {X₃₉₂₄₀,X₃₉₂₄₁}-harmonic conjugate of polar conjugate of X₁₈₈₇₉