X₁₂₇₇

3rd EVANS PERSPECTOR

X₁₂₇₇ = X₁₅-of-excentral-triangle

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

X₁₂₇₇ (3rd EVANS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 9}, {1, 16}

• Related Centers: X₄ (ORTHOCENTER), X₉ (MITTENPUNKT), X₁ (INCENTER), X₁₆ (2nd ISODYNAMIC POINT)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sqrt{3/4} - \sin(A - \pi /3) + \sin(B - \pi /3) + \sin(C - \pi /3)\)

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

Search Value (6, 9, 13): \(7.192644236262\)

▸ Alternate Coordinates1 form

• Trilinears: \(- \sqrt{3/4} - \cos(A + \pi /6) + \cos(B + \pi /6) + \cos(C + \pi /6)\)

X₁₂₇₇ lies on the Neuberg cubic.

▸ Incident Central Lines4 lines

{1, 16}, {4, 9}, {13, 484}, {63, 617}

▸ Functional Relations4 relations

• X₁₂₇₇ = reflection of X₁₂₇₆ in X₅₀₁₁

• X₁₂₇₇ = anticomplement of X₃₃₃₉₆

• X₁₂₇₇ = X₁₅-of-excentral-triangle

• X₁₂₇₇ = inverse-in-Bevan-circle of X₁₂₇₆ (noted by Peter J. C. Moses, Sept. 8, 2004)

Notes

Continuing from X₁₂₇₆, the triangles T and T″ are perspective with perspector X₁₂₇₇. (Lawrence Evans, 2/4/2003).

X₁₂₇₇ is the perspector of the excentral triangle and the apices of equilateral triangles constructed inward from the sides, as in the construction of X₁₄; for a generalization, see X₁₂₇₆. (Paul Yiu, 2/27/04).