X₁₃₃₅

{X₁,X₆}-HARMONIC CONJUGATE OF X₁₁₂₄

X₁₃₃₅ is the {X₁,X₆}-harmonic conjugate of X₁₁₂₄.

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

X₁₃₃₅ ({X₁,X₆}-HARMONIC CONJUGATE OF X₁₁₂₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {55, 371}

• Related Centers: X₁₃₃₆ (ISOGONAL CONJUGATE OF X₁₃₃₅), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₅₅ (INSIMILICENTER), X₃₇₁ (KENMOTU POINT)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 1 - \sin A\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(3.232789786235\)

▸ Alternate Coordinates1 form

• Trilinears: \(a(S - bc) : b(S - ca) : c(S - ab)\)

▸ Incident Central Lines13 lines

{1, 6}, {11, 486}, {12, 485}, {35, 1151}, {36, 1152}, {42, 493}, {55, 371}, {56, 372}, {81, 1123}, {175, 651}, {255, 606}, {498, 590}, {499, 615}

▸ Functional Relations4 relations

• X₁₃₃₅ = isogonal conjugate of X₁₃₃₆

• X₁₃₃₅ = isotomic conjugate of polar conjugate of X₃₄₁₂₁

• X₁₃₃₅ = X₁₉-isoconjugate of X₁₃₃₈₆

• X₁₃₃₅ = exsimilicenter of incircle and 2nd Lemoine circle; the insimilicenter is X₁₁₂₄