X₁₁₉₃

3rd SARAGOSSA POINT OF X₁

X₁₁₉₃ = crossdifference of every pair of points on line X₅₂₂X₆₄₉

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

X₁₁₉₃ (3rd SARAGOSSA POINT OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3293}

• Related Centers: X₁ (INCENTER), X₃₂₉₃ (SS(a → b + c) OF X₃)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{2} + c^{2} + ab + ac)\) (M. Iliev, 5/13/2007)

Search Value (6, 9, 13): \(1.119060399675\)

▸ Alternate Coordinates1 form

• Trilinears: \(a(as + S_{A}) : b(bs + S_{B}) : c(cs + S_{C})\) (C. Lozada, 9/07/2013)

▸ Incident Central Lines20 lines

{1, 2}, {3, 31}, {6, 41}, {21, 238}, {35, 595}, {36, 58}, {37, 992}, {38, 72}, {39, 213}, {57, 959}, {63, 988}, {106, 1126}, {171, 404}, {222, 1106}, {244, 942}, {405, 748}, {474, 750}, {518, 872}, {999, 1066}, {1045, 1050}

▸ Functional Relations8 relations

• X₁₁₉₃ = midpoint of X₁ and X₃₂₉₃

• X₁₁₉₃ = isogonal conjugate of X₁₂₂₀

• X₁₁₉₃ = isotomic conjugate of X₁₂₄₀

• X₁₁₉₃ = crosspoint of X_i and X_j for these (i,j): (1, 58), (6, 893), (57, 86)

• X₁₁₉₃ = crosssum of X_i and X_j for these (i,j): (1, 10), (2, 894), (9, 42)

• X₁₁₉₃ = crossdifference of every pair of points on line X₅₂₂X₆₄₉

• X₁₁₉₃ = X₉₂-isoconjugate of X₂₃₅₉

• X₁₁₉₃ = polar conjugate of isotomic conjugate of X₂₂₀₉₇

Notes

X₁₁₉₃ is the {X₁,X₄₃}-harmonic conjugate of X₈.