X₃₂₁₉

INTERSECTION OF LINES X₂X_7 AND X₈X_90

This point occurs in a forthcoming paper by Sadi Abu-Saymeh, Mowaffaq Hajja, and Hellmuth Stachel, “Another cubic associated with the triangle.”

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

X₃₂₁₉ (INTERSECTION OF LINES X₂X_7 AND X₈X_90) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 7}, {21, 72}

• Related Centers: X₂₁₆₀ (-ISOCONJUGATE OF X₃₅), X₂ (CENTROID), X₇ (GERGONNE POINT), X₂₁ (SCHIFFLER POINT), X₇₂ (ISOGONAL CONJUGATE OF X₂₈)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{2} - b^{2} - c^{2} - bc\)

Search Value (6, 9, 13): \(4.090281792597\)

▸ Alternate Coordinates1 form

• Trilinears: \((1 + 2 \cos A) \csc A : (1 + 2 \cos B) \csc B : (1 + 2 \cos C) \csc C\) (M. Iliev, 4/12/07)

▸ Incident Central Lines35 lines

{1, 2308}, {2, 7}, {8, 90}, {10, 191}, {21, 72}, {31, 984}, {37, 81}, {38, 238}, {42, 846}, {45, 940}, {55, 1776}, {71, 1654}, {100, 210}, {101, 1790}, {172, 593}, {190, 321}, {201, 1935}, {219, 1993}, {220, 394}, {281, 1748}, {306, 2895}, {312, 1150}, {323, 1442}, {324, 1948}, {612, 1707}, {651, 1214}, {748, 982}, {756, 896}, {799, 1920}, {912, 1006}, {938, 1728}, {958, 2099}, {960, 1319}, {1082, 2307}, {1268, 2160}

▸ Functional Relations9 relations

• X₃₂₁₉ = isogonal conjugate of X₂₁₆₀

• X₃₂₁₉ = isotomic conjugate of X₃₀₆₉₀

• X₃₂₁₉ = complement of X₁₇₄₈₃

• X₃₂₁₉ = X₁₂₆₈-Ceva conjugate of X₁

• X₃₂₁₉ = cevapoint of X₉ and X₁₉₁

• X₃₂₁₉ = X₃₅-cross conjugate of X₁₄₄₂

• X₃₂₁₉ = crosssum of X_i and X_j for these (i,j): (649, 3125), (1652, 1653)

• X₃₂₁₉ = crossdifference of every pair of points on the line X₆₆₃X₂₅₂₀

• X₃₂₁₉ = {X₉,X₅₇}-harmonic conjugate of X₃₃₀₅