X₁₁₅₈

CIRCUMCENTER OF EXTOUCH TRIANGLE

X₁₁₅₈ is the circumcenter of extouch triangle.

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

X₁₁₅₈ (CIRCUMCENTER OF EXTOUCH TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Extouch Triangle △N_aN_bN_c — vertices N_a, N_b, N_c (Contact triangle of Excircle touchpoints on BC, CA, AB)

• Circle: circumcircle_ofextouch — Centered at X₁₁₅₈

• Concurrence Lines: N_a–X₁₁₅₈, N_b–X₁₁₅₈, N_c–X₁₁₅₈ concurring at X₁₁₅₈

Trilinears: \(a^{6} - b^{6} - c^{6} + b^{2}c^{2}(b^{2} + c^{2}) + 3a^{2}(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2}) + 2abc(a^{3} - b^{3} - c^{3} - abc + (a^{2} + bc)(b + c) - ab^{2} - ac^{2})\)

Search Value (6, 9, 13): \(-20.447978971747\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sin^{2} B/2 \cos B + \sin^{2} C/2 \cos C - \sin^{2} A/2 \cos A\) (D. Grinberg, 2/25/04)

▸ Incident Central Lines8 lines

{1, 104}, {3, 960}, {4, 46}, {8, 20}, {57, 946}, {65, 1012}, {117, 208}, {165, 191}

▸ Functional Relations8 relations

• X₁₁₅₈ = midpoint of X₄₀ and X₈₄

• X₁₁₅₈ = complement of isotomic conjugate of X₃₄₄₁₃

• X₁₁₅₈ = X₃₁₈-Ceva conjugate of X₁

• X₁₁₅₈ = X₆₈-of-Fuhrmann-triangle

• X₁₁₅₈ = excentral isogonal conjugate of X₁₇₄₅

• X₁₁₅₈ = X₁₁₄₇-of-excentral-triangle

• X₁₁₅₈ = ABC-to-excentral barycentric image of X₃

• X₁₁₅₈ = X₃-of-extouch triangle, so that X₂₁₀X₁₁₅₈ = Euler line of the extouch triangle