X₂₀₇₆

INVERSE-IN-CIRCUMCIRCLE OF X₃₉

Let T₁ be the circumcevian triangle of the 1st Brocard point, and let T₂ be the circumcevian triangle of the 2nd Brocard point. Let V_AV_BV_C be the vertex triangle of T₁ and T₂, and let \(S_{A}\)\(S_{B}\)\(S_{C}\) be the side triangle of T₁ and T₂. The lines V_A\(S_{A}\), V_B\(S_{B}\), V_C\(S_{C}\) concur in X₂₀₇₆.

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

X₂₀₇₆ (INVERSE-IN-CIRCUMCIRCLE OF X₃₉) 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, 39}

• Related Centers: X₃ (CIRCUMCENTER), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sin A - 2 \cos A \sin 2\omega\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-2.785481148478\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a(a^{4} - b^{4} - c^{4} + a^{2}b^{2} + a^{2}c^{2} - b^{2}c^{2})\) (M. Iliev,5/13/07)

▸ Incident Central Lines9 lines

{3, 6}, {22, 1613}, {99, 732}, {141, 384}, {385, 698}, {599, 1003}, {691, 755}, {733, 805}, {904, 1964}

▸ Functional Relations12 relations

• X₂₀₇₆ = reflection of X_i in X_j for these (i,j): (6, 1691), (1691, 187)

• X₂₀₇₆ = inverse-in-circumcircle of X₃₉

• X₂₀₇₆ = X₆₉₄-Ceva conjugate of X₆

• X₂₀₇₆ = crosspoint of X₂₄₉ and X₈₀₅

• X₂₀₇₆ = crosssum of X₁₁₅ and X₈₀₄

• X₂₀₇₆ = reflection of X₁₆₉₁ in the Lemoine axis

• X₂₀₇₆ = radical center of Lucas(-csc 2ω) circles

• X₂₀₇₆ = intersection of tangents to circumcircle at intersections with line X₃₉X₅₁₂

• X₂₀₇₆ = crossdifference of every pair of points on line X₅₂₃X₃₅₈₉

• X₂₀₇₆ = {X₃,X₆}-harmonic conjugate of X₅₁₁₆

• X₂₀₇₆ = perspector of tangential triangle and the vertex triangle of the anticevian triangles of PU(1)

• X₂₀₇₆ = Brocard axis intercept, other than X₃, of circle {X₃,PU(2)}

Notes

Let T be the reflection of the symmedial triangle in the Lemoine axis; i.e., the reflection of the cevian triangle of X₆ in the trilinear polar of X₆. The triangle T is perspective to ABC, and the perspector is X(2076. (Randy Hutson, February 16, 2015)