X₂₉

CEVAPOINT OF INCENTER AND ORTHOCENTER

X₂₉ is the cevapoint of incenter and orthocenter.

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

X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER) 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, 92}

• Related Centers: X₁ (INCENTER), X₄ (ORTHOCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

Trilinears: \((\sec A)/(\cos B + \cos C) : (\sec B)/(\cos C + \cos A) : (\sec C)/(\cos A + \cos B)\)

Barycentrics: \((\tan A)/(\cos B + \cos C) : (\tan B)/(\cos C + \cos A) : (\tan C)/(\cos A + \cos B)\)

Search Value (6, 9, 13): \(5.278261169382\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a - b - c)/((b + c) (a^{2} - b^{2} - c^{2})) : :\)

▸ Incident Central Lines32 lines

{1, 92}, {2, 3}, {8, 219}, {10, 1794}, {33, 78}, {34, 77}, {58, 162}, {65, 296}, {81, 189}, {102, 107}, {112, 1311}, {226, 951}, {242, 257}, {270, 283}, {284, 950}, {314, 1039}, {388, 1037}, {392, 1871}, {497, 1036}, {515, 947}, {648, 1121}, {662, 1800}, {758, 1844}, {894, 1868}, {960, 1859}, {1056, 1059}, {1057, 1058}, {1125, 1838}, {1220, 1474}, {1737, 1780}, {1807, 1897}, {1842, 1848}

▸ Functional Relations18 relations

• X₂₉ = isogonal conjugate of X₇₃

• X₂₉ = isotomic conjugate of X₃₀₇

• X₂₉ = inverse-in-circumcircle of X₂₀₇₅

• X₂₉ = complement of X₃₁₅₂

• X₂₉ = anticomplement of X₁₈₆₄₁

• X₂₉ = X₂₈₆-Ceva conjugate of X₂₇

• X₂₉ = cevapoint of X_i and X_j for these (i,j): (1, 4), (33, 281)

• X₂₉ = X_i-cross conjugate of X_j for these (i,j): (1, 21), (284, 333), (497, 314)

• X₂₉ = crosssum of X_i and X_j for these (i,j): (1, 1047), (228, 1409)

• X₂₉ = crossdifference of every pair of points on line X₆₄₇X₈₂₂

• X₂₉ = X₄-Hirst inverse of X₄₁₅

• X₂₉ = X_i-beth conjugate of X_j for these (i,j): (29, 28), (811, 29)

• X₂₉ = intersection of tangents at X₁ and X₄ to hyperbola passing through X₁, X₄ and the excenters

• X₂₉ = crosspoint of X₁ and X₄ wrt the excentral triangle

• X₂₉ = trilinear pole of line X₂₄₃X₅₂₂ (the polar of X₂₂₆ wrt polar circle)

• X₂₉ = pole wrt polar circle of trilinear polar of X₂₂₆ (line X₅₂₃X₆₅₆)

• X₂₉ = polar conjugate of X₂₂₆

• X₂₉ = X₆-isoconjugate of X₁₂₁₄

Notes

As a point on the Euler line, X₂₉ has Shinagawa coefficients \((F S^{2}, \$bcS_{B}S_{C}\$ - F S^{2})\).

X₂₉ is the {X₃,X₄}-harmonic conjugate of X₄₁₂.