X₁₁₅₇

INVERSE-IN-CIRCUMCIRCLE OF X₅₄

For any point X, let X_A be the reflection of X in sideline BC, and define X_B and X_C cyclically. Then X₁₁₅₇ is the unique point X for which the lines AX_A, BX_B, CX_C concur on the circumcircle; the point of concurrence is X₁₁₄₁.

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

X₁₁₅₇ (INVERSE-IN-CIRCUMCIRCLE OF X₅₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 54}

• Related Centers: X₃ (CIRCUMCENTER), X₅₄ (KOSNITA POINT)

Trilinears: \(a[(a^{2} - b^{2})^{2} - c^{2}(a^{2} + b^{2})][(a^{2} - c^{2})^{2} - b^{2}(c^{2} + a^{2})]U(a,b,c)\), where \(U(a,b,c) = a^{6} - b^{6} - c^{6} + 3a^{2}(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2}) + b^{2}c^{2}(b^{2} + c^{2}) - a^{2}b^{2}c^{2}\)

Search Value (6, 9, 13): \(4.612380768446\)

▸ Alternate Coordinates1 form

• Trilinears: \(4 \cos A + \cos 3A \sec A \sec(B - C) : :\)

X₁₁₅₇ lies on the Neuberg cubic.

▸ Incident Central Lines9 lines

{1, 3483}, {3, 54}, {4, 3482}, {5, 252}, {30, 1141}, {74, 3484}, {186, 933}, {1337, 1338}, {3065, 3465}

▸ Functional Relations6 relations

• X₁₁₅₇ = isogonal conjugate of X₁₂₆₃

• X₁₁₅₇ = inverse-in-circumcircle of X₅₄

• X₁₁₅₇ = X₃₀-Ceva conjugate of X₃₄₈₄

• X₁₁₅₇ = Yiu-isogonal conjugate of X₁₉₅

• X₁₁₅₇ = Cundy-Parry Phi transform of X₁₉₅

• X₁₁₅₇ = Cundy-Parry Psi transform of X₃₄₅₉

Notes

X₁₁₅₇ is the tangential of X₃ on the Neuberg cubic.

Let Ref_aRef_bRef_c be the reflection triangle. The circumcircles of ARef_bRef_c, BRef_cRef_a, CRef_aRef_b (i.e., the Yiu circles) concur in X₁₁₅₇. (Randy Hutson, July 20, 2016)