X₁₉₃₇

TRILINEAR POLE OF PU(15)

X₁₉₃₇ = trilinear pole of line X₆₅X₆₅₀

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

X₁₉₃₇ (TRILINEAR POLE OF PU(15)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {21, 73}, {1, 185}

• Related Centers: X₁₉₃₆ (CROSSDIFFERENCE OF PU(15)), X₂₁ (SCHIFFLER POINT), X₇₃ (CROSSPOINT OF INCENTER AND CIRCUMCENTER), X₁ (INCENTER), X₁₈₅ (NAGEL POINT OF THE ORTHIC TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = 1/(\cos^{2} A - \cos B \cos C)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-3.000079575118\)

▸ Incident Central Lines13 lines

{1, 185}, {4, 774}, {8, 201}, {21, 73}, {80, 1736}, {90, 1745}, {104, 1458}, {108, 1172}, {225, 1896}, {307, 314}, {515, 1694}, {851, 1758}, {885, 1769}

▸ Functional Relations6 relations

• X₁₉₃₇ = isogonal conjugate of X₁₉₃₆

• X₁₉₃₇ = cevapoint of X_i and X_j for these (i,j): (1, 1758), (243, 1940)

• X₁₉₃₇ = trilinear pole of line X₆₅X₆₅₀

• X₁₉₃₇ = pole wrt polar circle of trilinear polar of X₁₉₄₈

• X₁₉₃₇ = X₄₈-isoconjugate (polar conjugate) of X₁₉₄₈

• X₁₉₃₇ = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the MacBeath circumconic