X₃₆₄₈

KS(ANTICOMPLEMENTARYTRIANGLE)

X₃₆₄₈ = anticomplement of X₇₉

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

X₃₆₄₈ (KS(ANTICOMPLEMENTARYTRIANGLE)) 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, 79}

• Related Centers: X₂ (CENTROID), X₇₉ (ISOGONAL CONJUGATE OF X₃₅)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[(2 + 3R/r)S^{2} - (ac + 2S_{B})(ab + 2S_{C})]\)

Search Value (6, 9, 13): \(7.794277765509\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B, C, A) : g(C, A, B)\), where \(g(A,B,C) = (\csc A)[(2/R + 3/r) \operatorname{area}(ABC) - (\sin A)(1 + 2 \cos B)(1 + 2 \cos C)] (f\) and g from Wimalasiri Perera, 11/16/2011)

• Trilinears: \(h(a,b,c) : h(b,c,a) : h(c,a,b)\), where \(h(a,b,c) = bc[3a^{4} + 2a^{3}(b + c) - a^{2}(2b^{2} + 2c^{2} + bc) - 2a(b^{3} + c^{3}) - abc(b + c) - (b^{2} - c^{2})^{2}]\) (Seiichi Kirikami, 12/13/2011)

▸ Linear Combinations1 combination

• X₃₆₄₈ = \(3\) X₂ - \(2\) X₇₉ = \((r + 4R)\) X₇ - \(2(r + 3R)\) X₂₁

▸ Incident Central Lines8 lines

{2, 79}, {7, 21}, {8, 30}, {10, 191}, {40, 153}, {63, 2894}, {145, 758}, {149, 3065}

▸ Functional Relations1 relation

• X₃₆₄₈ = anticomplement of X₇₉

Notes

Let O_A be the circle centered at the A-vertex of the 2nd Fuhrmann triangle and passing through A; define O_B and O_C cyclically. X₃₆₄₈ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)

Let O_A be the circle centered at the A-vertex of the K798e triangle and passing through A; define O_B and O_C cyclically. X₃₆₄₈ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)