X₁₃₁₆

5th MOSES INTERSECTION

X₁₃₁₆ = X₂₄₅₂ + 2 X₂₄₅₃

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

X₁₃₁₆ (5th MOSES INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 2453}, {2, 3} (Euler line)

• Related Centers: X₆ (SYMMEDIAN POINT), X₂₄₅₃ (REFLECTION OF X₆ IN X₁₃₁₆), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(bc \sin^{2}(B - C) + a^{2} \sin(A - B) \sin(A - C) : :\)

Barycentrics: \(a^{8} + a^{4}b^{2}c^{2} - a^{6}(b^{2} + c^{2}) + b^{2}c^{2}(b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(1.675519962766\)

▸ Alternate Coordinates1 form

• Barycentrics: \(b^{2}c^{2}(b^{2} - c^{2})^{2} + a^{4}(a^{2} - b^{2})(a^{2} - c^{2})\)

X₁₃₁₆ lies on the Brocard circle, the orthosymmedial circle, the cubics K023, K166, K508, K509, K876, K1091.

▸ Incident Central Lines77 lines

{2, 3}, {6, 523}, {51, 31850}, {76, 17941}, {98, 5191}, {99, 9155}, {110, 2782}, {115, 8429}, {125, 2794}, {182, 6795}, {183, 34245}, {184, 18338}, {247, 5972}, {250, 264}, {262, 842}, {287, 10753}, {323, 32515}, {338, 1576}, {373, 3111}, {476, 2698}, {538, 3292}, {542, 16280}, {543, 5465}, {691, 3972}, {1083, 24271}, {1561, 2777}, {1632, 20975}, {1648, 7737}, {1649, 14685}, {1968, 34859}, {1975, 2396}, {2394, 9168}, {2549, 16319}, {2790, 5622}, {3233, 35259}, {3447, 3613}, {3734, 5108}, {3815, 16320}, {3849, 32225}, {5091, 24288}, {5099, 5475}, {5149, 13518}, {5468, 6090}, {5476, 16279}, {5489, 11123}, {6232, 12508}, {6322, 11594}, {6531, 9475}, {6792, 18307}, {6794, 15048}, {7668, 7669}, {7698, 9159}, {7735, 16315}, {7736, 16316}, {7816, 11052}, {8723, 11183}, {8754, 23583}, {9169, 15539}, {11130, 14185}, {11131, 14187}, {11422, 14480}, {11586, 16632}, {12079, 26869}, {13434, 15112}, {14265, 32545}, {14682, 32314}, {15033, 15111}, {15743, 16633}, {16321, 32113}, {19128, 32428}, {19571, 25332}, {22505, 30789}, {22512, 30465}, {22513, 30468}, {23635, 30716}, {30715, 34845}, {32222, 32424}

▸ Functional Relations35 relations

• X₁₃₁₆ = X₂₄₅₂ + 2 X₂₄₅₃

• X₁₃₁₆ = midpoint of X₆ and X₂₄₅₃

• X₁₃₁₆ = reflection of X_i in X_j for these {i,j}: {2452, 6}, {5112, 468}, {6795, 182}, {16279, 5476}, {32113, 16321}, {32224, 32217}

• X₁₃₁₆ = isogonal conjugate of X₉₅₁₃

• X₁₃₁₆ = complement of X₃₆₁₆₃

• X₁₃₁₆ = anticomplement of X₁₁₀₀₇

• X₁₃₁₆ = crosspoint of X₁₆₀₇₀ and X₁₆₀₇₁

• X₁₃₁₆ = crosssum of X₁₃₄₁₄ and X₁₃₄₁₅

• X₁₃₁₆ = crossdifference of PU(145)

• X₁₃₁₆ = antigonal conjugate of X₃₈₉₄₇

• X₁₃₁₆ = X₂-Ceva conjugate of X₃₉₀₇₈

• X₁₃₁₆ = perspector of conic {{A, B, C,X₉₈,X₆₄₈}}

• X₁₃₁₆ = circumcircle-inverse of X₂₃₇

• X₁₃₁₆ = nine-point-circle-inverse of X₂₄₅₀

• X₁₃₁₆ = 2nd-Lemoine-circle-inverse of X₂₄₅₁

• X₁₃₁₆ = orthocentroidal-circle-inverse of X₈₆₈

• X₁₃₁₆ = polar-circle-inverse of X₂₉₇

• X₁₃₁₆ = orthoptic-circle-of-Steiner-inellipse-inverse of X₁₅₁₃

• X₁₃₁₆ = circumcircle-of-inner-Napoleon-triangle-inverse of X₁₀₈₀

• X₁₃₁₆ = circumcircle-of-outer-Napoleon-triangle-inverse-of X₃₈₃

• X₁₃₁₆ = psi-transform of X₉₈

• X₁₃₁₆ = X₁-isoconjugate of X₉₅₁₃

• X₁₃₁₆ = crosspoint of X₁₆₀₇₀ and X₁₆₀₇₁

• X₁₃₁₆ = crosssum of X₁₃₄₁₄ and X₁₃₄₁₅

• X₁₃₁₆ = crossdifference of every pair of points on line X₅₁₁X₆₄₇

• X₁₃₁₆ = intersection, other than X₆, of Brocard circles of ABC and orthocentroidal triangle

• X₁₃₁₆ = X₆-Hirst inverse of X₅₂₃

• X₁₃₁₆ = X₁₁₀ of 1st Brocard triangle

• X₁₃₁₆ = orthocentroidal-to-1st-Brocard similarity image of X₂

• X₁₃₁₆ = crosssum of S1 and S2 on the Brocard (third) cubic, K019, these being the Brocard-circle intercepts of the line X₂X₉₈

• X₁₃₁₆ = 1st-Brocard-isogonal conjugate of X₆₉₀

• X₁₃₁₆ = 1st-Brocard-isotomic conjugate of X₃₅₆₉

• X₁₃₁₆ = barycentric product X₂₉₆₆*X₃₁₉₅₃

• X₁₃₁₆ = barycentric quotient X_i/X_j for these {i,j}: {6, 9513}, {31953, 2799}

• X₁₃₁₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 868}, {2, 20, 35922}, {2, 23, 9832}, {2, 4226, 3}, {2, 10684, 14960}, {2, 26255, 14694}, {3, 25, 21525}, {4, 2409, 25}, {98, 35278, 5191}, {338, 1576, 9512}, {401, 419, 237}, {441, 460, 2450}, {868, 15000, 2}, {1113, 1114, 237}, {1312, 1313, 2450}, {5004, 5005, 15915}, {16179, 16180, 381}, {16181, 16182, 3}, {32460, 32461, 2}, {35606, 35906, 1640}

Notes

As a point on the Euler line, X₁₃₁₆ has Shinagawa coefficients \((3(E + F)F - S^{2}, -(E + F)^{2} + 3S^{2})\).

X₁₃₁₆ is the point of intersection, other than X₃, of the Euler line and Brocard circle.

X₁₃₁₆ is the point of intersection, other than X₆, of the Brocard circle and orthosymmedial circle.

X₁₃₁₆ is the point of intersection of the Euler line and the trilinear polar of X₉₈. (P.J.C. Moses, 6/22/04)

X₁₃₁₆ is the orthogonal projection of X₆ on the Euler line.

Let A₂B₂C₂, A₃B₃C₃, A₄B₄C₄ be the 2nd, 3rd and 4th Brocard triangles, respectively. Let A' = B₃B₄∩C₃C₄, and define B' and C' cyclically. The lines A₂A', B₂B', C₂C' concur in X₁₃₁₆. (Randy Hutson, January 29, 2015)