X₁₃₇₉

1st BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE

The Brocard axis, OK, is the line of the circumcenter, O [= X₃], and the symmedian point, K [= X₆]. This line meets the circumcircle in two points, X₁₃₇₉ and X₁₃₈₀; the closer to X₆ is X(1379.

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

X₁₃₇₉ (1st 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, 1348}

• Related Centers: X₃₄₁₃ (1st KIEPERT INFINITY POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₃₁₆ (DROUSSENT PIVOT), X₁₃₄₈ (INSIMILICENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a + 2(|OK| - R)\cot \omega \cos A\)

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

Search Value (6, 9, 13): \(-0.283321301067\)

▸ Alternate Coordinates2 forms

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

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

• X₁₃₇₉ = \((-1 + |OK|/R)\) X₃ + X₆

▸ Incident Central Lines23 lines

{2, 2039}, {3, 6}, {4, 2040}, {20, 35913}, {23, 6141}, {30, 31863}, {83, 14633}, {98, 3414}, {99, 3413}, {110, 5638}, {111, 5639}, {141, 19660}, {230, 31862}, {316, 1348}, {352, 6142}, {476, 13722}, {513, 36735}, {517, 36736}, {620, 14502}, {667, 11651}, {1113, 35607}, {1114, 14899}, {1349, 38227}

▸ Functional Relations13 relations

• X₁₃₇₉ = isogonal conjugate of X₃₄₁₃

• X₁₃₇₉ = reflection of X_i in X_j for these (i,j): (4, 2040), (1380, 3), (3557, 2029)

• X₁₃₇₉ = anticomplement of X₂₀₃₉

• X₁₃₇₉ = inverse-in-Brocard-circle of X₁₃₄₁

• X₁₃₇₉ = X₂₄₉-Ceva conjugate of X₁₃₈₀

• X₁₃₇₉ = Ψ(X₂, X₁₃₄₀)

• X₁₃₇₉ = trilinear pole of line X₆X₅₆₃₉

• X₁₃₇₉ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₃₅₅₈

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

• X₁₃₇₉ = circumcircle intercept, other than A, B, C, of hyperbola {A, B, C,X₆,PU(118)}

• X₁₃₇₉ = perspector of triangles AiBiCi and (A_aB_bC_c)*, and of triangles A_aB_bC_c and (AiBiCi)*; see preamble before X₁₁₇₅₂

• X₁₃₇₉ = perspector of ABC and (degenerate) cross-triangle of 1st and 2nd anti-circummedial triangles

• X₁₃₇₉ = barycentric product of circumcircle intercepts of line X₂X₁₃₄₀