X₁₉₁₄

CROSSDIFFERENCE OF PU(10)

X₁₉₁₄ = intersection of trilinear polars of P(9) and U(9)

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

X₁₉₁₄ (CROSSDIFFERENCE OF PU(10)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 32}, {6, 31}

• Related Centers: X₃₃₅ (ISOTOMIC CONJUGATE OF X₂₃₉), X₁ (INCENTER), X₃₂ (3rd POWER POINT), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT)

Trilinears: \(1 - a^{2}/(bc) : :\)

Search Value (6, 9, 13): \(-1.372253800065\)

▸ Alternate Coordinates1 form

• Trilinears: \(a^{3} - abc : :\)

▸ Incident Central Lines29 lines

{1, 32}, {6, 31}, {9, 983}, {11, 230}, {21, 1107}, {35, 39}, {36, 187}, {37, 82}, {44, 765}, {48, 1613}, {81, 593}, {100, 1575}, {105, 910}, {112, 1870}, {213, 595}, {284, 893}, {292, 1438}, {350, 385}, {577, 1040}, {584, 1185}, {604, 1403}, {649, 834}, {727, 813}, {739, 901}, {741, 1326}, {999, 1384}, {1055, 1149}, {1319, 1415}, {1428, 1691}

▸ Functional Relations13 relations

• X₁₉₁₄ = isogonal conjugate of X₃₃₅

• X₁₉₁₄ = isotomic conjugate of X₁₈₈₉₅

• X₁₉₁₄ = complement of X₂₀₅₅₃

• X₁₉₁₄ = anticomplement of X₂₀₅₄₁

• X₁₉₁₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 39029), (727, 31), (1429, 1428), (1438, 6), (1911, 172)

• X₁₉₁₄ = crosspoint of X_i and X_j for these (i,j): (81, 105), (238, 1429), (239, 242), (904, 1911), (919, 1252)

• X₁₉₁₄ = crosssum of X_i and X_j for these (i,j): (37, 518), (292, 295), (350, 1909), (918, 1086)

• X₁₉₁₄ = {X₁,X₃₂}-harmonic conjugate of X₁₇₂

• X₁₉₁₄ = intersection of trilinear polars of P(9) and U(9)

• X₁₉₁₄ = X₉₂-isoconjugate of X₂₉₅

• X₁₉₁₄ = perspector of hyperbola {{A, B, C,X₅₈,X₁₀₁,PU(9)}}

• X₁₉₁₄ = barycentric product of PU(134)

• X₁₉₁₄ = homothetic center of intangents triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

Notes

X₁₉₁₄ is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(9) and U(9) of bicentric points (see the notes just before X₁₉₀₈. (Randy Hutson, 9/23/2011)