X₁₃₂₅

CIRCUMCIRCLE-INVERSE OF X₂₁

X₁₃₂₅ is the circumcircle-inverse of X₂₁.

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

X₁₃₂₅ (CIRCUMCIRCLE-INVERSE OF X₂₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1290, 36}, {2, 3} (Euler line)

• Related Centers: X₁₂₉₀ (COLLINGS TRANSFORM OF X₃₆), X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = [b^{4} + c^{4} - a^{4} + abc(b + c - a) - 2b^{2}c^{2}]/(b + c)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-4.548104700147\)

▸ Incident Central Lines8 lines

{1, 229}, {2, 3}, {36, 759}, {60, 65}, {104, 476}, {105, 691}, {110, 517}, {1295, 1304}

▸ Functional Relations6 relations

• X₁₃₂₅ = reflection of X₁₂₉₀ in X₃₆

• X₁₃₂₅ = circumcircle-inverse of X₂₁

• X₁₃₂₅ = nine-point-circle-inverse of X₃₇₉₈₃

• X₁₃₂₅ = crosspoint of X₃ and X₂₉₄₈ wrt both the excentral and tangential triangles

• X₁₃₂₅ = crossdifference of every pair of points on line X₆₄₇X₂₀₉₂

• X₁₃₂₅ = reflection of X₁₁₀ in line X₃₆X₂₃₈ (the polar of X₁ wrt circumcircle)

Notes

As a point on the Euler line, X₁₃₂₅ has Shinagawa coefficients \((\$a\$(E + 4F) + 2abc, -4\$a\$(E + F) - 6abc)\).