X₃₈₈

X₁X_4∩X₇X_8

X₃₈₈ = perspector of ABC and 5th extouch triangle

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

X₃₈₈ (X₁X_4∩X₇X_8) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 958}

• Related Centers: X₂ (CENTROID), X₉₅₈ (INTERSECTION OF LINES X)

Trilinears: \(bc[a^{2} + (b + c)^{2}]/(b + c - a)\)

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

Search Value (6, 9, 13): \(0.936658594328\)

▸ Alternate Coordinates1 form

• Trilinears: \(1 + \cos B \cos C : 1 + \cos C \cos A : 1 + \cos A \cos B\)

▸ Linear Combinations1 combination

• X₃₈₈ = \(2(R/r)\) X₁ + \(3\) X₂ - \(2\) X₃

▸ Incident Central Lines24 lines

{1, 4}, {2, 12}, {3, 495}, {5, 999}, {7, 8}, {10, 57}, {11, 153}, {20, 55}, {29, 1037}, {35, 376}, {36, 498}, {79, 1000}, {108, 406}, {171, 603}, {201, 984}, {329, 960}, {354, 938}, {355, 942}, {381, 496}, {442, 956}, {452, 1001}, {612, 1038}, {750, 1106}, {1059, 1067}

▸ Functional Relations9 relations

• X₃₈₈ = isogonal conjugate of X₁₀₃₆

• X₃₈₈ = anticomplement of X₉₅₈

• X₃₈₈ = X₁₁₈₁-of-intouch triangle

• X₃₈₈ = perspector of ABC and 5th extouch triangle

• X₃₈₈ = homothetic center of anticomplementary triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

• X₃₈₈ = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

• X₃₈₈ = isotomic conjugate of X₃₀₄₇₉

• X₃₈₈ = homothetic center of anti-inverse-in-incircle triangle and anti-tangential midarc triangle

• X₃₈₈ = centroid of curvatures of anticomplementary circle and excircles

Notes

Let A″B″C″ be the reflection of the Mandart-incircle triangle in X₁. A″B″C″ is homothetic to the anticomplementary triangle at X₃₈₈. (Randy Hutson, September 14, 2016)

X₃₈₈ is the {X₇,X₈}-harmonic conjugate of X₆₅.