X₁₉₇₇

BICENTRIC DIFFERENCE OF PU(42)

X₁₉₇₇ = trilinear pole wrt symmedial triangle of line X₁X₆

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

X₁₉₇₇ (BICENTRIC DIFFERENCE OF PU(42)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 100}, {213, 1017}

• Related Centers: X₆ (SYMMEDIAN POINT), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₂₁₃ (-CEVA CONJUGATE OF X₄₂), X₁₀₁₇ (POINT ALFIRK)

Trilinears: \(a^{3}(b - c)^{2} : :\)

Search Value (6, 9, 13): \(0.272943217699\)

X₁₉₇₇ lies on the Brocard inellipse.

▸ Incident Central Lines4 lines

{6, 100}, {213, 1017}, {291, 1017}, {1397, 1501}

▸ Functional Relations8 relations

• X₁₉₇₇ = isogonal conjugate of X₃₁₆₂₅

• X₁₉₇₇ = X_i-Ceva conjugate of X_j for these (i,j): (6, 667), (31, 669), (32, 1919), (739, 890), (1397, 1980)

• X₁₉₇₇ = crosspoint of X_i and X_j for these (i,j): (6, 667), (32, 1919), (87, 1019)

• X₁₉₇₇ = crosssum of X_i and X_j for these (i,j): (2, 668), (43, 1018), (76, 1978)

• X₁₉₇₇ = trilinear pole wrt symmedial triangle of line X₁X₆

• X₁₉₇₇ = polar conjugate of isotomic conjugate of X₂₂₀₉₆

• X₁₉₇₇ = crossdifference of every pair of points on line X₆₆₈X₈₉₁ (the tangent to the Steiner circumellipse at X₆₆₈)

• X₁₉₇₇ = barycentric square of X₆₄₉