X₃₀₆₃

CROSSDIFFERENCE OF X₇ AND X₈

X₃₀₆₃ is the perspector of the triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X₅₅, and X₅₆.

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

X₃₀₆₃ (CROSSDIFFERENCE OF X₇ AND X₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {6, 513}, {59, 919}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₅₁₃ (ISOGONAL CONJUGATE OF X₁₀₀), X₅₉ (ISOGONAL CONJUGATE OF X₁₁), X₉₁₉ (ISOGONAL CONJUGATE OF X₉₁₈)

Trilinears: \(a^{2}(b - c)(b + c - a) : b^{2}(c - a)(c + a - b) : c^{2}(a - b)(a + b - c)\)

Barycentrics: \(a^{3}(b - c)(b + c - a) : b^{3}(c - a)(c + a - b) : c^{3}(a - b)(a + b - c)\)

Search Value (6, 9, 13): \(-4.539010263157\)

▸ Incident Central Lines10 lines

{6, 513}, {59, 919}, {112, 2714}, {521, 650}, {649, 854}, {657, 663}, {661, 832}, {665, 1459}, {667, 788}, {834, 2483}

▸ Functional Relations10 relations

• X₃₀₆₃ = X_i-Ceva conjugate of X_j for these (i,j): (2, 39025), (644, 55), (649, 667), (692, 2175), (919, 2223), (1415, 31), (2192, 3022), (2423, 1960)

• X₃₀₆₃ = isogonal conjugate of X₄₅₅₄

• X₃₀₆₃ = cevapoint of X₇₉₈ and X₃₀₄₉

• X₃₀₆₃ = crosspoint of X_i and X_j for these (i,j): (6, 692), (31, 1415), (55, 644), (649, 663), (651, 1037)

• X₃₀₆₃ = crosssum of X_i and X_j for these (i,j): (2, 693), (37, 2533), (190, 664), (497, 650), (513, 2275)

• X₃₀₆₃ = crossdifference of every pair of points on the line X₇X₈

• X₃₀₆₃ = intersection of trilinear polars of X₅₅ and X₅₆

• X₃₀₆₃ = trilinear product of PU(103)

• X₃₀₆₃ = perspector of hyperbola {{A, B, C,X₅₅,X₅₆}}

• X₃₀₆₃ = barycentric product of Feuerbach hyperbola intercepts of Lemoine axis