X₂₂₉₂

X₂-ISOCONJUGATE OF X₁₁₆₉

X₂₂₉₂ = anticomplement wrt incentral triangle of X₁

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

X₂₂₉₂ (X₂-ISOCONJUGATE OF X₁₁₆₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2650, 1}

• Related Centers: X₂₆₅₀ (SUM OF PU(81)), X₁ (INCENTER)

Trilinears: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\), where f(a,b,c) is as in X₁₁₆₉

Barycentrics: \(a^{2}/f(a,b,c) : b^{2}/f(b,c,a): c^{2}/f(c,a,b)\)

Search Value (6, 9, 13): \(2.265606493377\)

▸ Incident Central Lines24 lines

{1, 21}, {2, 986}, {8, 192}, {10, 321}, {29, 240}, {37, 65}, {40, 612}, {42, 72}, {46, 750}, {55, 976}, {190, 1220}, {221, 2256}, {226, 1254}, {244, 1125}, {392, 1201}, {690, 2254}, {950, 2310}, {960, 1193}, {1042, 1214}, {1107, 2170}, {1717, 1773}, {1761, 2303}, {1829, 2269}, {1868, 1880}

▸ Functional Relations9 relations

• X₂₂₉₂ = isogonal conjugate of X₂₃₆₃

• X₂₂₉₂ = reflection of X₂₆₅₀ in X₁

• X₂₂₉₂ = complement of X₁₇₁₆₄

• X₂₂₉₂ = X_i-Ceva conjugate of X_j for these (i,j): (1, 1193), (109, 656), (190, 661)

• X₂₂₉₂ = crosspoint of X_i and X_j for these (i,j): (1, 10), (75, 226)

• X₂₂₉₂ = crosssum of X_i and X_j for these (i,j): (1, 58), (31, 284)

• X₂₂₉₂ = complement wrt incentral triangle of X₂₆₅₀

• X₂₂₉₂ = anticomplement wrt incentral triangle of X₁

• X₂₂₉₂ = incentral isotomic conjugate of X₂₆₆₇

Notes

Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. The triangle A'B'C' is perspective to the incentral triangle at X₂₂₉₂. (Randy Hutson, December 2, 2017)

Let F_aF_bF_c be the Fuhrmann triangle, and A_bB_cC_a and A_cB_aC_b be the 1st and 2nd Montesdeoca bisector triangles. X₂₂₉₂ is the radical center of the circumcircles of F_aA_bA_c, F_bB_cB_a, F_cC_aC_b. (Randy Hutson, December 2, 2017)