X₁₁₁₃

1st EULER-LINE-CIRCUMCIRCLE INTERSECTION

X₁₁₁₃ is a point of intersection of the Euler line and the circumcircle. The other is X₁₁₁₄. Of the two, X₁₁₁₃ is the one closer to X₄.

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

X₁₁₁₃ (1st EULER-LINE-CIRCUMCIRCLE INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {109, 1822}

• Related Centers: X₂₅₇₄ (ISOGONAL CONJUGATE OF X₁₁₁₃), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₁₈₂₂ (ISOGONAL CONJUGATE OF X₂₅₈₈)

Trilinears: \((R - d)\cos A - 2R \cos B \cos C : (R - d)\cos B - 2R \cos C \cos A : (R - d)\cos C - 2R \cos A \cos B\), where R = circumradius, d = distance |OH| between X₃ and X₄. (Joe Goggins, 2002)

Barycentrics: \(2RS_{B}S_{C} + (|OH| - R)a^{2}S_{A} : :\), where |OH| = distance between X₃ and X₄, and R = circumradius (Peter J. C. Moses, 3/2003; cf. X₁₃₁₃, X₁₃₁₄)

Search Value (6, 9, 13): \(0.391075321794\)

▸ Alternate Coordinates2 forms

• Trilinears: \((1 - J) \cos A - 2 \cos B \cos C : :\), where \(J = |OH|/R = (1/abc)[S(6) - S(2,4) + 3a^{2}b^{2}c^{2}]^{1/2}\), where \(S(6) = a^{6} + b^{6} + c^{6}\), and \(S(2,4) = a^{2}b^{4} + a^{2}c^{4} + b^{2}c^{4} + b^{2}a^{4} + c^{2}a^{4} + c^{2}b^{4}\) (Peter J. C. Moses, 10/2/03)

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) - a^{2} (-a^{2} + b^{2} + c^{2}) \cdot (1 - J) : :\)

▸ Linear Combinations1 combination

• X₁₁₁₃ = \(3\) X₂ + \((-3 + |OH|/R)\) X₃ = X₁₁₁₃ = \((- 1 + |OH|/R)\) X₃ + X₄

▸ Incident Central Lines26 lines

{1, 2100}, {2, 3}, {6, 2104}, {11, 10781}, {54, 14374}, {74, 2575}, {98, 2593}, {100, 2580}, {101, 2576}, {108, 2586}, {109, 1822}, {110, 2574}, {111, 8106}, {112, 8105}, {165, 2101}, {187, 8426}, {511, 2105}, {517, 2103}, {691, 9173}, {759, 2589}, {1495, 13415}, {2249, 2579}, {2777, 14500}, {5840, 10782}, {5972, 14499}, {10287, 10686}

▸ Functional Relations13 relations

• X₁₁₁₃ = reflection of X_i in X_j for these (i,j): (4, 1312), (1114, 3)

• X₁₁₁₃ = isogonal conjugate of X₂₅₇₄

• X₁₁₁₃ = isotomic conjugate of X₂₂₃₃₉

• X₁₁₁₃ = anticomplement of X₁₃₁₃

• X₁₁₁₃ = X₂₅₀-Ceva conjugate of X₁₁₁₄

• X₁₁₁₃ = trilinear product X₁₁₀*X₁₈₂₃

• X₁₁₁₃ = trilinear pole of line X₆X₁₃₄₅ (the major axis of the orthic inconic)

• X₁₁₁₃ = Ψ(X₆, X₁₃₄₅)

• X₁₁₁₃ = pole wrt polar circle of trilinear polar of X₂₅₉₂ (line X₅₂₃X₁₃₁₃)

• X₁₁₁₃ = polar conjugate of X₂₅₉₂

• X₁₁₁₃ = inverse-in-polar-circle of X₁₃₁₃

• X₁₁₁₃ = Thomson isogonal conjugate of X₂₅₇₅

• X₁₁₁₃ = Lucas isogonal conjugate of X₂₅₇₅

Notes

As a point on the Euler line, X₁₁₁₃ has Shinagawa coefficients \((R - |OH|, -3R + |OH|)\).

X₁₁₁₃ is one of 2 points P such that P is the circumcircle-antipode of Λ (trilinear polar of P); the other is X₁₁₁₄. (Randy Hutson, November 2, 2017)