X₁₃₀₄

COLLINGS TRANSFORM OF X₄₀₃

X₁₃₀₄ = X₁₈₆-cross conjugate of X₂₅₀

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

X₁₃₀₄ (COLLINGS TRANSFORM OF X₄₀₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2693, 3}

• Related Centers: X₂₆₉₃ (SR(X₇₄, X₁₀₇)), X₃ (CIRCUMCENTER)

Trilinears: \((\sin A)/[(\tan B - \tan C)(\tan B + \tan C - 2 \tan A)] : :\)

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

Search Value (6, 9, 13): \(0.547074900796\)

X₁₃₀₄ lies on the circumcircle.

▸ Incident Central Lines54 lines

{2, 2697}, {3, 2693}, {4, 477}, {21, 2694}, {23, 1297}, {25, 842}, {27, 2688}, {28, 2687}, {29, 2695}, {30, 1294}, {74, 186}, {98, 468}, {99, 3233}, {100, 5379}, {102, 2075}, {103, 2073}, {104, 2074}, {107, 523}, {110, 250}, {111, 232}, {112, 647}, {378, 841}, {403, 1300}, {476, 4240}, {648, 1302}, {691, 4230}, {877, 2855}, {925, 7471}, {933, 1624}, {935, 2394}, {1141, 5627}, {1290, 4246}, {1292, 7476}, {1295, 1325}, {1296, 7482}, {1305, 7479}, {1494, 2373}, {2071, 5897}, {2159, 2249}, {2328, 2738}, {2360, 2732}, {2370, 7478}, {2433, 2715}, {2689, 7452}, {2690, 4241}, {2691, 4238}, {2696, 4235}, {2734, 3109}, {2752, 4233}, {2758, 4248}, {2766, 7435}, {2770, 4232}, {3470, 3518}, {3565, 7468}

▸ Functional Relations22 relations

• X₁₃₀₄ = reflection of X₂₆₉₃ in X₃

• X₁₃₀₄ = isogonal conjugate of X₉₀₃₃

• X₁₃₀₄ = X₁₈₆-cross conjugate of X₂₅₀

• X₁₃₀₄ = cevapoint of X₄₀₃ and X₅₂₃

• X₁₃₀₄ = crosssum of X₃₀ and X₄₀₂

• X₁₃₀₄ = cevapoint of {403, 523}, {647, 1495}

• X₁₃₀₄ = X_i-cross conjugate of X_j for these (i,j): (186, 250), (526, 4), (686, 2052), (3003, 249),. (5502, 110)

• X₁₃₀₄ = trilinear pole of {6, 74}

• X₁₃₀₄ = trilinear product X_i*X_j for these {i,j}: {74, 162}, {112, 2349}, {648, 2159}, {662, 8749}

• X₁₃₀₄ = barycentric product X_i*X_j for these {i,j}: {74, 648}, {99, 8749}, {112, 1494}, {162, 2349}, {250, 2394}, {811, 2159}

• X₁₃₀₄ = polar-circle-inverse of X₃₂₅₈

• X₁₃₀₄ = Moses-radical-circle-inverse of X₁₁₂

• X₁₃₀₄ = trilinear pole of line X₆X₇₄

• X₁₃₀₄ = Ψ(X_i, X_j) for these (i,j): (3, 74), (6, 74), (69, 74)

• X₁₃₀₄ = Λ(X_i, X_j) for these (i,j): (74, 1294), (107, 110), (113, 133), (122, 125), (1636, 1637), (1494, 3268)

• X₁₃₀₄ = reflection of X₁₀₇ in the Euler line

• X₁₃₀₄ = inverse-in-polar-circle of X₃₂₅₈

• X₁₃₀₄ = inverse-in-Moses-radical-circle of X₁₁₂

• X₁₃₀₄ = isotomic conjugate of polar conjugate of X₃₂₆₉₅

• X₁₃₀₄ = polar conjugate of isogonal conjugate of X₃₂₆₄₀

• X₁₃₀₄ = perspector of circumorthic triangle and cross-triangle of ABC and circumcevian triangle of X₁₈₆

• X₁₃₀₄ = X_i-isoconjugate of X_j for these {i,j}: {1, 9033}, {2, 2631}, {30, 656}, {63, 1637}, {92, 1636}, {162, 1650}, {520, 1784}, {525, 2173}, {810, 3260}, {1568, 2616}, {1577, 3284}, {2407, 3708}, {2632, 4240}, {6357, 8611}

Notes

Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₃₀₄. (Randy Hutson, February 10, 2016)