X₁₁₉₄

3rd SARAGOSSA POINT OF X₂

X₁₁₉₄ = crossdifference of every pair of points on line X₅₂₅X₆₆₉

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

X₁₁₉₄ (3rd SARAGOSSA POINT OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 25}, {2, 39}

• Related Centers: X₁₂₄₁ (ISOTOMIC CONJUGATE OF X₁₁₉₄), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₂ (CENTROID), X₃₉ (BROCARD MIDPOINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{4} + c^{4} + a^{2}b^{2} + a^{2}c^{2})\) (M. Iliev, 5/13/07)

Search Value (6, 9, 13): \(1.250279411812\)

▸ Incident Central Lines5 lines

{2, 39}, {6, 25}, {22, 32}, {23, 251}, {230, 570}

▸ Functional Relations4 relations

• X₁₁₉₄ = isotomic conjugate of X₁₂₄₁

• X₁₁₉₄ = crosspoint of X_i and X_j for these (i,j): (2, 251), (4, 308), (6, 893)

• X₁₁₉₄ = crosssum of X_i and X_j for these (i,j): (2, 384), (6, 141)

• X₁₁₉₄ = crossdifference of every pair of points on line X₅₂₅X₆₆₉

Notes

Let L be the isogonal conjugate of the isotomic conjugate of line X₂X₆ (i.e., line X₆X₂₅). Let M be the isotomic conjugate of the isogonal conjugate of line X₂X₆ (i.e., line X₂X₃₉). Then X₁₁₉₄ = L∩M. (Randy Hutson, March 21, 2019)