X₂₃₈

X₁-LINE CONJUGATE OF X₃₇

X₂₃₈ = X₁-line conjugate of X₃₇

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

X₂₃₈ (X₁-LINE CONJUGATE OF X₃₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 1279}

• Related Centers: X₁ (INCENTER), X₁₂₇₉ (MIDPOINT OF X₁ AND X₂₃₈)

Trilinears: \(a^{2} - bc : b^{2} - ca : c^{2} - ab\)

Barycentrics: \(a^{3} - abc : b^{3} - abc : c^{3} - abc\)

Search Value (6, 9, 13): \(-3.700405115298\)

▸ Incident Central Lines49 lines

{1, 6}, {2, 31}, {3, 978}, {4, 602}, {7, 1471}, {8, 983}, {10, 82}, {21, 256}, {36, 513}, {40, 1722}, {42, 1621}, {43, 55}, {47, 499}, {56, 87}, {57, 1707}, {58, 86}, {63, 614}, {71, 1244}, {100, 899}, {105, 291}, {106, 898}, {162, 415}, {190, 726}, {212, 497}, {239, 740}, {241, 1456}, {242, 419}, {244, 896}, {390, 1253}, {459, 1395}, {484, 1739}, {516, 673}, {517, 1052}, {519, 765}, {580, 946}, {601, 631}, {651, 1458}, {662, 1326}, {942, 1046}, {987, 1472}, {992, 1009}, {993, 995}, {1006, 1064}, {1040, 1711}, {1054, 1155}, {1284, 1428}, {1465, 1758}, {1479, 1714}, {1699, 1754}

▸ Functional Relations21 relations

• X₂₃₈ = midpoint of X₁ and X₁₂₇₉

• X₂₃₈ = reflection of X₁ in X₁₂₉₇

• X₂₃₈ = isogonal conjugate of X₂₉₁

• X₂₃₈ = isotomic conjugate of X₃₃₄

• X₂₃₈ = anticomplement of X₃₈₃₆

• X₂₃₈ = X_i-Ceva conjugate of X_j for these (i,j): (105, 1), (292, 171)

• X₂₃₈ = X₆₅₉-cross conjugate of X₃₅₇₃

• X₂₃₈ = crosssum of X_i and X_j for these (i,j): (10, 726), (42, 672), (239, 894)

• X₂₃₈ = crossdifference of every pair of points on line X₃₇X₅₁₃

• X₂₃₈ = X_i-Hirst inverse of X_j for these (i,j): (1, 6), (43, 55)

• X₂₃₈ = X₁-line conjugate of X₃₇

• X₂₃₈ = X₁₀₅-aleph conjugate of X₂₃₈

• X₂₃₈ = X_i-beth conjugate of X_j for these (i,j): (21, 238), (643, 902), (644, 238), (932, 238)

• X₂₃₈ = {X₁,X₉}-harmonic conjugate of X₉₈₄

• X₂₃₈ = intersection of trilinear polars of PU(8)

• X₂₃₈ = inverse-in-circumconic-centered-at-X₉ of X₆

• X₂₃₈ = crossdifference of PU(i) for these i: 6, 52, 53

• X₂₃₈ = trilinear product of PU(134)

• X₂₃₈ = X₆₅₃₀-of-excentral-triangle

• X₂₃₈ = trilinear pole of line X₆₅₉X₄₄₃₅ (the perspectrix of ABC and Gemini triangle 33)

• X₂₃₈ = perspector of Gemini triangle 34 and cross-triangle of ABC and Gemini triangle 34

Notes

X₂₃₈ is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(8) and U(8) of bicentric points (see the notes just before X₁₉₀₈. (Randy Hutson, 9/23/2011)