X₉₃₄

X₅₁₃-CROSS CONJUGATE OF X₅₇

X₉₃₄ = X₅₁₃-cross conjugate of X₅₇

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

X₉₃₄ (X₅₁₃-CROSS CONJUGATE OF X₅₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {972, 3}

• Related Centers: X₉₇₂ (ISOGONAL CONJUGATE OF X₉₇₁), X₃ (CIRCUMCENTER)

Trilinears: \(1/(bw - cv) : :\), where \(u : v : w\) = X₅₇

Search Value (6, 9, 13): \(0.042941416030\)

▸ Alternate Coordinates1 form

• Trilinears: \(1/[(b - c)(b + c - a)^{2}] : :\)

X₉₃₄ lies on the circumcircle.

▸ Incident Central Lines13 lines

{1, 103}, {3, 972}, {7, 104}, {56, 105}, {77, 102}, {100, 658}, {101, 651}, {106, 269}, {644, 1025}, {675, 1088}, {727, 1106}, {741, 1042}, {759, 1014}

▸ Functional Relations20 relations

• X₉₃₄ = reflection of X₉₇₂ in X₃

• X₉₃₄ = isogonal conjugate of X₃₉₀₀

• X₉₃₄ = isotomic conjugate of X₄₃₉₇

• X₉₃₄ = X₅₁₃-cross conjugate of X₅₇

• X₉₃₄ = perspector of ABC and triangle formed by line X₃X₉ reflected in sides of ABC

• X₉₃₄ = concurrence of reflections in sides of ABC of line X₄X₇

• X₉₃₄ = Λ(X₆₅₀, X₆₆₃) (Λ(trilinear polar of X₉)

• X₉₃₄ = Λ(X₃₀₆₄, X₃₇₀₀) (Λ(trilinear polar of X₂₈₁)

• X₉₃₄ = trilinear pole of line X₆X₅₇

• X₉₃₄ = polar-circle-inverse of X₃₈₉₆₆

• X₉₃₄ = X₂-Ceva conjugate of X₂₃₉₇₁

• X₉₃₄ = Ψ(X₁, X₇)

• X₉₃₄ = Ψ(X₃, X₇₇)

• X₉₃₄ = Ψ(X₄, X₇)

• X₉₃₄ = Ψ(X₆, X₅₇)

• X₉₃₄ = Ψ(X₉, X₅₇)

• X₉₃₄ = trilinear pole, wrt circumtangential triangle, of line X₃X₉

• X₉₃₄ = trilinear product of intercepts of circumcircle and Soddy line

• X₉₃₄ = eigencenter of 4th mixtilinear triangle

• X₉₃₄ = intersection of antipedal lines of circumcircle intercepts of line X₃X₉

Notes

Let P be a point on line X₄X₇ other than X₄. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur at X₉₃₄. (Randy Hutson, July 20, 2016)