X₁₃₈₀

2nd BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE

X₁₃₈₀ = inverse-in-Brocard-circle of X₁₃₄₀

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

X₁₃₈₀ (2nd BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {316, 1349}

• Related Centers: X₃₄₁₄ (2nd KIEPERT INFINITY POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₃₁₆ (DROUSSENT PIVOT), X₁₃₄₉ (EXSIMILICENTER)

Trilinears: \(a - 2(|OK| + R)\cot \omega \cos A : :\)

Search Value (6, 9, 13): \(13.848047089460\)

▸ Alternate Coordinates2 forms

• Trilinears: \(e \cos A + \cos(A + \omega ) : :\), where \(e = (1 - 4 \sin^{2}\omega )^{1/2}\)

• Trilinears: \(a(a^{4} - b^{2}c^{2} - a^{2}T) : :\), where \(T = (a^{4} + b^{4} + c^{4} - b^{2}c^{2} - c^{2}a^{2} - a^{2} b^{2})^{1/2}\)

▸ Linear Combinations1 combination

• X₁₃₈₀ = \((1 + |OK|/R)\) X₃ - X₆

▸ Incident Central Lines23 lines

{2, 2040}, {3, 6}, {4, 2039}, {20, 35914}, {23, 6142}, {30, 31862}, {83, 14632}, {98, 3413}, {99, 3414}, {110, 5639}, {111, 5638}, {141, 19659}, {230, 31863}, {316, 1349}, {352, 6141}, {476, 13636}, {513, 36736}, {517, 36735}, {620, 14501}, {667, 11652}, {1113, 35608}, {1114, 35609}, {1348, 38227}

▸ Functional Relations10 relations

• X₁₃₈₀ = isogonal conjugate of X₃₄₁₄

• X₁₃₈₀ = reflection of X_i in X_j for these (i,j): (4, 2039), (1379, 3), (3558, 2028)

• X₁₃₈₀ = anticomplement of X₂₀₄₀

• X₁₃₈₀ = inverse-in-Brocard-circle of X₁₃₄₀

• X₁₃₈₀ = X₂₄₉-Ceva conjugate of X₁₃₇₉

• X₁₃₈₀ = Ψ(X₂, X₁₃₄₁)

• X₁₃₈₀ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₃₅₅₇

• X₁₃₈₀ = {X₁₆₈₇,X₁₆₈₈}-harmonic conjugate of X₁₃₇₉

• X₁₃₈₀ = trilinear pole of line X₆X₅₆₃₈ (tangent to hyperbola {{A, B, C,X₆,PU(118)}} at X₆)

• X₁₃₈₀ = barycentric product of circumcircle intercepts of line X₂X₁₃₄₁

Notes

Let A' be the incenter of BCX₁₅, and define B' and C' cyclically. Let A″ be the incenter of BCX₁₆, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₁₃₈₀. (Randy Hutson, February 10, 2016)

In the plane of a triangle ABC, let

• F₁ = X₃₉₁₆₂ and F₂ = X₃₉₁₆₃; these points are the foci of the Steiner inellipse;

• L₁ = trilinear polar of F₁

• L₂ = trilinear polar of F₁

• A₁ = L₁∩BC, and define B₁ and C₁ cyclically

• A₂ = L₂∩BC, and define B₂ and C₂ cyclically

The seven circle (ABC), (AB₁C₁), (BC₁A₁), (CA₁B₁), (AB₂C₂), (BC₂A₂), (CA₁B₂) concur in X₁₃₈₀. See X₁₃₈₀ (X(1380).png) (Angel Montesdeoca, February 18, 2023)