X₃₈₄₃

X₃₀₉₁com(EULER TRIANGLE)

X₃₈₄₃ = homothetic center of Ehrmann vertex-triangle and anti-incircle-circles triangle

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

X₃₈₄₃ (X₃₀₉₁com(EULER TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1159, 1837}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₁₅₉ (GREENHILL POINT), X₁₈₃₇ (ZOSMA TRANSFORM OF X₃₄)

Barycentrics: \(3a^{4} - 4b^{4} - 4c^{4} + a^{2}b^{2} + a^{2}c^{2} + 8b^{2}c^{2} : :\)

Search Value (6, 9, 13): \(-3.184822949902\)

▸ Incident Central Lines11 lines

{2, 3}, {68, 3531}, {265, 3527}, {355, 3625}, {946, 3635}, {999, 3585}, {1159, 1837}, {1352, 3630}, {1482, 1699}, {3295, 3583}, {3426, 3521}

▸ Functional Relations6 relations

• X₃₈₄₃ = complement of X₁₇₅₃₈

• X₃₈₄₃ = anticomplement of X₁₅₇₁₂

• X₃₈₄₃ = homothetic center of Ehrmann vertex-triangle and anti-incircle-circles triangle

• X₃₈₄₃ = homothetic center of Ehrmann mid-triangle and X3-ABC reflections triangle

• X₃₈₄₃ = endo-homothetic center of Ehrmann mid-triangle and anti-Euler triangle; the homothetic center is X₃₀₉₁

• X₃₈₄₃ = {X₂₀₄₃,X₂₀₄₄}-harmonic conjugate of X₅₄₇

Notes

As a point on the Euler line, X₃₈₄₃ has Shinagawa coefficients \((1, 7)\).

Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). Let A'B'C' be the Euler-of-Euler triangle. X₃₈₄₃ is the radical center of the circumcircles of triangles A’AeAi, B’BeBi, C’CeCi. (Randy Hutson, June 7, 2019)