X₁₁₁₄

2nd EULER-LINE-CIRCUMCIRCLE INTERSECTION

X₁₁₁₄ is a point of intersection of the Euler line and the circumcircle. Its antipode is X₁₁₁₃.

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

X₁₁₁₄ (2nd 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, 1823}

• Related Centers: X₂₅₇₅ (ISOGONAL CONJUGATE OF X₁₁₁₄), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₁₈₂₃ (ISOGONAL CONJUGATE OF X₂₅₈₉)

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 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)

Search Value (6, 9, 13): \(13.173650466599\)

▸ Alternate Coordinates2 forms

• Trilinears: \((1 + J) \cos A - 2 \cos B \cos C\), where \(J = |OH|/R\); see X₁₁₁₃

• 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

• X₁₁₁₄ = \(3\) X₂ - \((3 + |OH|/R)\) X₃ = \(- (1 + |OH|/R)\) X₃ + X₄

▸ Incident Central Lines27 lines

{1, 2101}, {2, 3}, {6, 2105}, {11, 10782}, {54, 14375}, {74, 2574}, {98, 2592}, {100, 2581}, {101, 2577}, {108, 2587}, {109, 1823}, {110, 2575}, {111, 8105}, {112, 8106}, {165, 2100}, {187, 8427}, {511, 2104}, {517, 2102}, {691, 9174}, {759, 2588}, {1379, 14899}, {1495, 13414}, {2249, 2578}, {2777, 14499}, {5840, 10781}, {5972, 14500}, {10288, 10687}

▸ Functional Relations13 relations

• X₁₁₁₄ = reflection of X_i in X_j for these (i,j): (4, 1313), (1113, 3)

• X₁₁₁₄ = isogonal conjugate of X₂₅₇₅

• X₁₁₁₄ = isotomic conjugate of X₂₂₃₄₀

• X₁₁₁₄ = anticomplement of X₁₃₁₂

• X₁₁₁₄ = X₂₅₀-Ceva conjugate of X₁₁₁₃

• X₁₁₁₄ = trilinear product X₁₁₀*X₁₈₂₂

• X₁₁₁₄ = trilinear pole of line X₆X₁₃₄₄ (the minor axis of the orthic inconic)

• X₁₁₁₄ = Ψ(X₆, X₁₃₄₄)

• X₁₁₁₄ = pole wrt polar circle of trilinear polar of X₂₅₉₃ (line X₅₂₃X₁₃₁₂)

• X₁₁₁₄ = polar conjugate of X₂₅₉₃

• X₁₁₁₄ = inverse-in-polar-circle of X₁₃₁₂

• X₁₁₁₄ = Thomson isogonal conjugate of X₂₅₇₄

• X₁₁₁₄ = Lucas isogonal conjugate of X₂₅₇₄

Notes

As a point on the Euler line, X₁₁₁₄ has Shinagawa coefficients \((R + |OH|, -3R - |OH|)\).

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