X₁₉₇₄

BARYCENTRIC PRODUCT OF PU(39)

X₁₉₇₄ = X₂₅-Ceva conjugate of X₃₂

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

X₁₉₇₄ (BARYCENTRIC PRODUCT OF PU(39)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 25}, {4, 83}

• Related Centers: X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₃₂ (3rd POWER POINT), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₈₃ (CEVAPOINT OF CENTROID AND SYMMEDIAN POINT)

Trilinears: \(a^{4}\cos B \cos C : :\)

Search Value (6, 9, 13): \(-0.097435557748\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\tan A \sin^{3} A : :\)

• Trilinears: \(\tan A \sin(A - \omega ) : :\)

▸ Incident Central Lines22 lines

{4, 83}, {6, 25}, {24, 511}, {32, 682}, {34, 1428}, {53, 460}, {66, 125}, {69, 459}, {110, 193}, {112, 729}, {141, 468}, {156, 1353}, {235, 1503}, {237, 577}, {264, 419}, {428, 597}, {571, 1576}, {981, 1172}, {1147, 1351}, {1386, 1829}, {1395, 1397}, {1691, 1968}

▸ Functional Relations9 relations

• X₁₉₇₄ = isogonal conjugate of X₃₀₅

• X₁₉₇₄ = X₂₅-Ceva conjugate of X₃₂

• X₁₉₇₄ = crosspoint of X_i and X_j for these {i,j}: {6, 34207}, {1395, 1973}

• X₁₉₇₄ = crosssum of X_i and X_j for these (i,j): (2, 1370), (304, 3718), (339, 3267)

• X₁₉₇₄ = crossdifference of every pair of points on the line X₅₂₅X₃₂₆₇

• X₁₉₇₄ = X₉₂-isoconjugate of X₃₉₂₆

• X₁₉₇₄ = trilinear product of vertices of Ara triangle

• X₁₉₇₄ = polar conjugate of X₁₅₀₂

• X₁₉₇₄ = barycentric product of intersections of circumcircle and 2nd Lemoine circle

Notes

X₁₉₇₄ is the X_i-isoconjugate of X_j for these (i,j): (48,1502), (92,3926); also X₁₉₇₄ is the pole with respect to the polar cirle of the trilinear polar of X₁₅₀₂. Randy Hutson, August 15, 2013