X₂₇₂₀

SR(X₁₀₀, X₁₀₈)

X₂₇₂₀ = cevapoint of X₆₆₃ and X₁₄₀₄

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

X₂₇₂₀ (SR(X₁₀₀, X₁₀₈)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2745, 3}

• Related Centers: X₂₇₄₅ (SR(X₁₀₄, X₁₀₈)), X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where f(a,b,c) is as given just before X₂₆₇₇ using P = X₁₀₀, U = X₁₀₈

Search Value (6, 9, 13): \(-0.157899240325\)

X₂₇₂₀ lies on the circumcircle.

▸ Incident Central Lines29 lines

{1, 2716}, {3, 2745}, {7, 2861}, {36, 102}, {56, 953}, {57, 2717}, {59, 100}, {65, 2687}, {101, 652}, {103, 2078}, {104, 1319}, {106, 1457}, {108, 513}, {109, 1459}, {497, 2723}, {517, 1395}, {840, 1617}, {883, 2865}, {909, 2272}, {915, 1870}, {919, 2423}, {929, 2401}, {944, 2734}, {972, 1155}, {1214, 2747}, {1397, 2726}, {1402, 2699}, {1420, 2718}, {2283, 2742}

▸ Functional Relations14 relations

• X₂₇₂₀ = reflection of X₂₇₄₅ in X₃

• X₂₇₂₀ = isogonal conjugate of X₂₈₀₄

• X₂₇₂₀ = cevapoint of X₆₆₃ and X₁₄₀₄

• X₂₇₂₀ = X_i-cross conjugate of X_j for these (i,j): (654, 57), (1319, 59)

• X₂₇₂₀ = concurrence of reflections in sides of ABC of line X₄X₁₁

• X₂₇₂₀ = trilinear pole of line X₆X₉₀₉

• X₂₇₂₀ = Ψ(X₄, X₁₁)

• X₂₇₂₀ = Ψ(X₆, X₉₀₉)

• X₂₇₂₀ = Ψ(X₉, X₁₀₁)

• X₂₇₂₀ = Ψ(X₆₅₀, X₆)

• X₂₇₂₀ = isotomic conjugate of polar conjugate of X₃₂₇₀₂

• X₂₇₂₀ = reflection of X₁₀₈ in line X₁X₃

• X₂₇₂₀ = inverse-in-circle-O(3513,3514)-of-X₁₀₅

• X₂₇₂₀ = perspector of 3rd mixtilinear triangle and cross-triangle of ABC and circumcevian triangle of X₁₃₁₉

Notes

Let A', B', C' be the intersections of line X₁X₃ and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₂₇₂₀. (Randy Hutson, December 10, 2016)

The tangent lines from X₂₇₂₀ to the incircle intersect the circumcircle in points on the Sherman Line, so forming a poristic triangle with ABC. (Peter Moses, December 3, 2019)