X₃₆₂₈

H⁻¹(X₅; M, 2)

For another construction see Antreas Hatzipolakis and Peter Moses, Euclid 2576.

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

X₃₆₂₈ (H⁻¹(X₅; M, 2)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 140}, {2, 3} (Euler line)

• Related Centers: X₅ (NINE-POINT CENTER), X₁₄₀ (MIDPOINT OF X₃ AND X₅), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(bc[a^{2}(-5b^{2} - 5c^{2} + 2a^{2}) + 3(b^{2} - c^{2})^{2}] : :\)

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

Search Value (6, 9, 13): \(2.110244529775\)

▸ Alternate Coordinates2 forms

• Trilinears: \(5 \cos A + 6 \cos B \cos C : :\)

• Trilinears: \(6 \sec A + 5 \sec B \sec C : :\)

▸ Linear Combinations1 combination

• X₃₆₂₈ = \(3\) X₂ + X₅ = X₃ + \(3\) X₅

▸ Incident Central Lines20 lines

{2, 3}, {17, 395}, {18, 396}, {32, 3054}, {39, 3055}, {52, 373}, {141, 576}, {143, 1216}, {156, 182}, {230, 1506}, {233, 3284}, {485, 3594}, {486, 3592}, {495, 499}, {496, 498}, {575, 3564}, {623, 630}, {624, 629}, {952, 1125}, {1209, 1493}

▸ Functional Relations15 relations

• X₃₆₂₈ = midpoint of X₅ and X₁₄₀

• X₃₆₂₈ = isogonal conjugate of X₃₄₅₆₇

• X₃₆₂₈ = isotomic conjugate of isogonal conjugate of X₃₄₅₆₅

• X₃₆₂₈ = complement of X₁₄₀

• X₃₆₂₈ = centroid of {A, B, C,X₅}

• X₃₆₂₈ = circumcircle-inverse of X₃₇₉₅₆

• X₃₆₂₈ = center of the Vu pedal-centroidal circle of X₁₄₀

• X₃₆₂₈ = Kosnita(X₅,X₂) point

• X₃₆₂₈ = center of conic which is locus of centers of circumconics passing through X₅

• X₃₆₂₈ = center of bicevian conic of X₂ and X₅

• X₃₆₂₈ = {X₂,X₅}-harmonic conjugate of X₁₄₀

• X₃₆₂₈ = {X₃,X₅}-harmonic conjugate of X₅₄₆

• X₃₆₂₈ = homothetic center of X₂-altimedial and X₁₄₀-anti-altimedial triangles

• X₃₆₂₈ = inverse-in-orthocentroidal-circle of X₃₅₂₆

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

Notes

As a point on the Euler line, X₃₆₂₈ has Shinagawa coefficients \((5, 1)\).

X₃₆₂₈ is the centroid of the set {M_a, M_b, M_c, X₅}, where M_aM_bM_c is the medial triangle; more generally, \(H^{-1}(X; M, 2)\) is the centroid of the set {M_a, M_b, M_c, X}. (Angel Montesdeoca, 12/20/2011)