X₃₁₆₇

X₆-CEVA CONJUGATE OF X₃

X₃₁₆₇ = Thomson-isogonal conjugate of X₂₀

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

X₃₁₆₇ (X₆-CEVA CONJUGATE OF X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 49}, {25, 110}

• Related Centers: X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₆ and \(u : v : w\) = X₃

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\), where \(f(a,b,c) = u(-u/x + v/y + w/z)\)

Search Value (6, 9, 13): \(2.084574017866\)

▸ Incident Central Lines16 lines

{3, 49}, {6, 1196}, {22, 323}, {25, 110}, {68, 1656}, {154, 511}, {193, 459}, {195, 973}, {392, 912}, {542, 1853}, {999, 1201}, {1200, 2280}, {1362, 1397}, {1384, 1501}, {1611, 1692}, {1994, 1995}

▸ Functional Relations8 relations

• X₃₁₆₇ = isogonal conjugate of X₃₄₂₀₈

• X₃₁₆₇ = crossdifference of every pair of points on line X₂₅₀₁X₃₅₆₆

• X₃₁₆₇ =X_i-Ceva conjugate of X_j for these (i,j): (6, 3), (193, 3053)

• X₃₁₆₇ = crosspoint of X_i and X_j for these {i,j}: {6, 3053}, {193, 6337}

• X₃₁₆₇ = crosssum of X_i and X_j for these (i,j): {2, 2996}, {523, 8754}, {2501, 5139}, {8770, 14248}

• X₃₁₆₇ = Thomson-isogonal conjugate of X₂₀

• X₃₁₆₇ = perspector of unary cofactor triangles of outer and inner Vecten triangles

• X₃₁₆₇ = centroid of anticevian triangle of X₃, which is also the antipedal triangle of X₆₄ and the tangential triangle of the MacBeath circumconic