X₃₅₂₆

EULER LINE INTERCEPT OF LINE X₄₉X_182

X₃₅₂₆ = midpoint of X₃₀₉₀ and X₃₅₂₃

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

X₃₅₂₆ (EULER LINE INTERCEPT OF LINE X₄₉X_182) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3090, 3523}, {2, 3} (Euler line)

• Related Centers: X₃₀₉₀ (INTERSECTION OF LINES X), X₃₅₂₃ (EULER LINE INTERCEPT OF LINE X), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 5 \cos A + 4 \cos B \cos C\)

Barycentrics: \(\cot B \cot C - 5 : :\)

Search Value (6, 9, 13): \(3.222653664161\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = 4 \sec A + 5 \sec B \sec C\)

▸ Incident Central Lines19 lines

{2, 3}, {6, 3411}, {49, 182}, {143, 2979}, {195, 394}, {302, 628}, {303, 627}, {498, 999}, {499, 3295}, {567, 1092}, {568, 1216}, {575, 599}, {590, 3312}, {615, 3311}, {1125, 1482}, {1159, 1788}, {1384, 2548}, {1385, 1698}, {1506, 3053}

▸ Functional Relations9 relations

• X₃₅₂₆ = midpoint of X₃₀₉₀ and X₃₅₂₃

• X₃₅₂₆ = reflection of X₃ in X₃₅₂₃

• X₃₅₂₆ = complement of X₃₀₉₀

• X₃₅₂₆ = circumcircle-inverse of X₃₇₉₄₇

• X₃₅₂₆ = orthocentroidal-circle-inverse of X₃₆₂₈

• X₃₅₂₆ = homothetic center of medial triangle and mid-triangle of Euler and anti-Euler triangles

• X₃₅₂₆ = homothetic center of X₅-altimedial and X₂-anti-altimedial triangles

• X₃₅₂₆ = {X₃₄₁₁,X₃₄₁₂}-harmonic conjugate of X₆

• X₃₅₂₆ = {X₂,X₄}-harmonic conjugate of X₃₆₂₈

Notes

As a point on the Euler line, X₃₅₂₆ has Shinagawa coefficients \((5, -1)\).

Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). Let A'B'C' be the medial-of-medial triangle. X₃₅₂₆ is the radical center of the circumcircles of triangles A’AeAi, B’BeBi, C’CeCi. (Randy Hutson, June 7, 2019)