X₁₉₇₈

TRILINEAR PRODUCT OF PU(41)

As the trilinear product of Steiner circumellipse antipodes, X₁₉₇₈ lies on conic {{A, B, C,X₆₆₈,X₇₈₉}} with center X₆₃₇₆ and perspector X₇₅. (Randy Hutson, July 11, 2019)

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

X₁₉₇₈ (TRILINEAR PRODUCT OF PU(41)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {75, 244}, {310, 321}

• Related Centers: X₁₉₁₉ (BICENTRIC DIFFERENCE OF PU(12)), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₂₄₄ (-LINE CONJUGATE OF X₁₀₀), X₃₁₀ (ISOTOMIC CONJUGATE OF X₄₂), X₃₂₁ (ISOTOMIC CONJUGATE OF X₈₁)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = b^{3}c^{3}/(b - c)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(7.657943703919\)

▸ Incident Central Lines9 lines

{75, 244}, {99, 835}, {100, 789}, {101, 689}, {190, 670}, {310, 321}, {312, 561}, {668, 891}, {811, 1897}

▸ Functional Relations9 relations

• X₁₉₇₈ = isogonal conjugate of X₁₉₁₉

• X₁₉₇₈ = isotomic conjugate of X₆₄₉

• X₁₉₇₈ = complement of X₂₁₂₂₄

• X₁₉₇₈ = anticomplement of X₆₃₇₇

• X₁₉₇₈ = X₆₇₀-Ceva conjugate of X₆₆₈

• X₁₉₇₈ = cevapoint of X_i and X_j for these (i,j): (75, 514), (321, 693), (646, 668), (850, 1230)

• X₁₉₇₈ = crosssum of X₆₆₉ and X₁₉₂₄

• X₁₉₇₈ = trilinear pole of line X₁₀X₇₅ (the isotomic conjugate of the isogonal conjugate of the Nagel line)

• X₁₉₇₈ = trilinear product of intercepts of Steiner circumellipse and line X₂X₃₇