X₂₈

CEVAPOINT OF X₁₉ AND X₂₅

X₂₈ is the cevapoint of X₁₉ and X₂₅.

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

X₂₈ (CEVAPOINT OF X₁₉ AND X₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1, 19}

• Related Centers: X₁₉ (CLAWSON POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁ (INCENTER)

Trilinears: \((\tan A)/(b + c) : (\tan B)/(c + a) : (\tan C)/(a + b)\)

Barycentrics: \((\sin A \tan A)/(b + c) : (\sin B \tan B)/(c + a) : (\sin C \tan C)/(a + b)\)

Search Value (6, 9, 13): \(-0.868703140076\)

▸ Linear Combinations1 combination

• X₂₈ = \(8 sa sb sc\) X₄ + \(a b c (3 + J^{2})\) X₂₅

▸ Incident Central Lines47 lines

{1, 19}, {2, 3}, {10, 1891}, {11, 1852}, {33, 975}, {34, 57}, {35, 1869}, {36, 1838}, {46, 1780}, {54, 1243}, {56, 278}, {60, 81}, {65, 1175}, {72, 1257}, {88, 162}, {104, 107}, {105, 112}, {108, 225}, {110, 915}, {142, 1890}, {228, 943}, {242, 261}, {272, 273}, {279, 1014}, {281, 958}, {291, 1783}, {501, 1831}, {579, 1724}, {580, 1730}, {607, 1002}, {608, 959}, {614, 1472}, {956, 1219}, {957, 1191}, {961, 1169}, {1104, 1333}, {1125, 1848}, {1155, 1888}, {1170, 1876}, {1178, 1432}, {1224, 1826}, {1255, 1824}, {1295, 1301}, {1385, 1871}, {1412, 1422}, {1633, 1770}, {1710, 1725}

▸ Functional Relations14 relations

• X₂₈ = isogonal conjugate of X₇₂

• X₂₈ = isotomic conjugate of X₂₀₃₃₆

• X₂₈ = anticomplement of X₂₁₅₃₀

• X₂₈ = trilinear pole of line X₅₁₃X₁₄₃₀ (the polar of X₃₂₁ wrt polar circle)

• X₂₈ = polar conjugate of X₃₂₁

• X₂₈ = X₆-isoconjugate of X₃₀₆

• X₂₈ = X₇₅-isoconjugate of X₂₂₈

• X₂₈ = circumcircle-inverse of X₂₀₇₄

• X₂₈ = X_i-Ceva conjugate of X_j for these (i,j): (270, 58), (286, 81)

• X₂₈ = cevapoint of X_i and X_j for these (i,j): (19, 25), (34, 56)

• X₂₈ = X_i-cross conjugate of X_j for these (i,j): (19, 27), (58, 58)

• X₂₈ = crossdifference of every pair of points on line X₆₄₇X₆₅₆

• X₂₈ = X₄-Hirst inverse of X₄₂₂

• X₂₈ = X_i-beth conjugate of X_j for these (i,j): (29, 29), (107, 28), (162, 28), (270, 28)

Notes

As a point on the Euler line, X₂₈ has Shinagawa coefficients \((\$a\$F, -\$a\$(E + F) - abc)\).

X₂₈ is the {X₂₇,X₂₉}-harmonic conjugate of X₄.