X₁₈₈₅

ZOSMA TRANSFORM OF X₇₇₄

X₁₈₈₅ = crosspoint of X₄ and X₁₁₀₅

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

X₁₈₈₅ (ZOSMA TRANSFORM OF X₇₇₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {389, 974}

• Related Centers: X₄ (ORTHOCENTER), X₁₁₀₅ (CEVAPOINT OF X₃ AND X₄), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₃₈₉ (CENTER OF THE TAYLOR CIRCLE), X₉₇₄ (2nd EHRMANN POINT)

Trilinears: (y + z) sec A : (z + x) sec B : (x + y) sec C,\(x : y : z\) = X₇₇₄

Barycentrics: \((2 a^{6} - 3 a^{4} (b^{2} + c^{2}) + 8 a^{2} b^{2} c^{2} + (b^{2} - c^{2})^{2} (b^{2} + c^{2})) / (b^{2} + c^{2} - a^{2}) : :\)

Search Value (6, 9, 13): \(-14.398038489980\)

▸ Incident Central Lines9 lines

{2, 3}, {64, 1899}, {389, 974}, {497, 1398}, {515, 1902}, {516, 1829}, {950, 1876}, {1039, 1721}, {1770, 1905}

▸ Functional Relations7 relations

• X₁₈₈₅ = anticomplement of X₃₁₈₂₉

• X₁₈₈₅ = crosspoint of X₄ and X₁₁₀₅

• X₁₈₈₅ = crosssum of X₃ and X₁₈₅

• X₁₈₈₅ = intersection of tangents to Hatzipolakis-Lozada hyperbola at X₄ and X₁₈₅

• X₁₈₈₅ = crosspoint, wrt orthic triangle, of X₄ and X₁₈₅

• X₁₈₈₅ = X₂₀-of-anti-Ara-triangle

• X₁₈₈₅ = X₃₀₅₇-of-orthic-triangle if ABC is acute

Notes

As a point on the Euler line, X₁₈₈₅ has Shinagawa coefficients \((F, E - 3F)\).

Let H_aH_bH_c be the orthic triangle. X₁₈₈₅ is the radical center of the 2nd Droz-Farny circles of triangles AH_bH_c, BH_cH_a, CH_aH_b. (Randy Hutson, July 31 2018)