X₁₅₈₅

POINT CAPH I

X₁₅₈₅ = inverse-in-orthocentroidal-circle of X₁₅₈₆

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

X₁₅₈₅ (POINT CAPH I) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {53, 590}

• Related Centers: X₁₅₉₉ (POINT CHORT I), X₁₉₉₃ (ORTHOCORRESPONDENT OF X₃), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE), X₅₉₀ (ISOGONAL CONJUGATE OF X₅₈₈)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \csc A + \sec A\)

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

Search Value (6, 9, 13): \(4.769602533119\)

▸ Incident Central Lines9 lines

{2, 3}, {53, 590}, {264, 491}, {275, 486}, {317, 492}, {343, 638}, {393, 493}, {394, 637}, {490, 1321}

▸ Functional Relations4 relations

• X₁₅₈₅ = inverse-in-orthocentroidal-circle of X₁₅₈₆

• X₁₅₈₅ = cevapoint of X₁₅₉₉ and X₁₉₉₃

• X₁₅₈₅ = X_i-cross conjugate of X_j for these (i,j): (371, 492), (1993, 1586)

• X₁₅₈₅ = polar conjugate of X₄₈₅

Notes

As a point on the Euler line, X₁₅₈₅ has Shinagawa coefficients \((F, S)\).