X₂₂₂

X₇-CEVA CONJUGATE OF X₅₆

X₂₂₂ = cevapoint of X₆ and X₂₂₁

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

X₂₂₂ (X₇-CEVA CONJUGATE OF X₅₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 27}, {6, 57}

• Related Centers: X₇ (GERGONNE POINT), X₅₆ (EXSIMILICENTER), X₂₇ (CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER), X₆ (SYMMEDIAN POINT), X₅₇ (ISOGONAL CONJUGATE OF X₉)

Trilinears: \(\cos A \tan A/2 : \cos B \tan B/2 : \cos C \tan C/2\)

Barycentrics: \(a^{2}/(1 + \sec A) : b^{2}/(1 + \sec B) : c^{2}/(1 + \sec C)\)

Search Value (6, 9, 13): \(-1.306049730322\)

▸ Alternate Coordinates2 forms

• Trilinears: \(1/(\csc A + 2 \csc 2A) : 1/(\csc B + 2 \csc 2B) : 1/(\csc A + 2 \csc 2C)\)

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

▸ Incident Central Lines23 lines

{1, 84}, {2, 651}, {3, 73}, {6, 57}, {7, 27}, {33, 971}, {34, 942}, {46, 227}, {55, 103}, {56, 58}, {63, 77}, {72, 1038}, {171, 611}, {189, 281}, {218, 241}, {226, 478}, {268, 1073}, {581, 1035}, {601, 1066}, {613, 982}, {912, 1060}, {1355, 1363}, {1993, 3218}

▸ Functional Relations16 relations

• X₂₂₂ = isogonal conjugate of X₂₈₁

• X₂₂₂ = isotomic conjugate of X₇₀₁₇

• X₂₂₂ = X_i-Ceva conjugate of X_j for these (i,j): (7, 56), (77, 3), (81, 57)

• X₂₂₂ = cevapoint of X₆ and X₂₂₁

• X₂₂₂ = X_i-cross conjugate of X_j for these (i,j): (48, 3), (73, 77)

• X₂₂₂ = crosspoint of X₇ and X₃₄₈

• X₂₂₂ = crosssum of X_i and X_j for these (i,j): (55, 607), (650, 1146)

• X₂₂₂ = crossdifference of every pair of points on line X₃₀₆₄X₃₇₀₀

• X₂₂₂ = trilinear pole of line X₁₄₅₉X₁₉₄₆

• X₂₂₂ = Danneels point of X₆₅₁ (see notes at X₃₀₇₈)

• X₂₂₂ = X₄-isoconjugate of X₉

• X₂₂₂ = intouch-isogonal conjugate of X₁₂₇₂₃

• X₂₂₂ = perspector of intouch triangle and unary cofactor triangle of extouch triangle

• X₂₂₂ = perspector of ABC and unary cofactor triangle of Gemini triangle 38

• X₂₂₂ = vertex conjugate of foci of inconic that is the isotomic conjugate of the polar conjugate of the incircle (centered at X₁₇₀₇₃)

• X₂₂₂ = X_i-beth conjugate of X_j for these (i,j): (21, 1012), (63, 63), (110, 222), (287, 222), (648, 222), (651, 222), (662, 2), (895, 222)

Notes

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X₂₂₂. See also X₆, X₂₅, X₂₁₈, X₉₄₀, X₁₇₄₃. (Randy Hutson, July 23, 2015)

Let T_aT_bT_c be the intouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line T_bT_c, and define B″ and C″ cyclically. The lines T_aA″, T_bB″, T_cC″ concur in X₂₂₂. (Randy Hutson, July 31 2018)