X₃₅₉₆

1st ODEHNAL POINT

X₃₅₉₆ = polar conjugate of X₆₀₈

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

X₃₅₉₆ (1st ODEHNAL POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {10, 75}, {69, 150}

• Related Centers: X₁₃₉₇ (-BETH CONJUGATE OF X₃₁), X₁₀ (SPIEKER CENTER), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₁₅₀ (REFLECTION OF X₂₀ IN X₁₀₃)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = b^{3}c^{3}(b + c - a)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(6.047283165386\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = \csc^{2} A \csc^{2}(A/2)\)

• Barycentrics: for the center of the circle A″ are given by Peter Moses (August 26, 2014):

▸ Incident Central Lines18 lines

{2, 1240}, {8, 314}, {10, 75}, {43, 350}, {69, 150}, {86, 996}, {92, 2064}, {190, 573}, {192, 2092}, {219, 645}, {261, 958}, {264, 1969}, {281, 345}, {304, 309}, {305, 561}, {312, 2321}, {346, 646}, {730, 1740}

▸ Functional Relations9 relations

• X₃₅₉₆ = isogonal conjugate of X₁₃₉₇

• X₃₅₉₆ = isotomic conjugate of X₅₆

• X₃₅₉₆ = isotomic conjugate of isogonal conjugate of X₈

• X₃₅₉₆ = isotomic conjugate of isogonal conjugate of anticomplement of X₁

• X₃₅₉₆ = anticomplement of X₁₇₀₅₃

• X₃₅₉₆ = complement of polar conjugate of isogonal conjugate of X₂₃₁₅₉

• X₃₅₉₆ = polar conjugate of X₆₀₈

• X₃₅₉₆ = trilinear product of extraversions of X₇

• X₃₅₉₆ = trilinear product of vertices of Gemini triangle 39

Notes

Let A', B', C' denote the respective excircles of a triangle ABC. Let A″ be the circle tangent to A', B', C' whose interior includes A', and likewise for circles B″ and C″. Let K_aa be the point of tangency of circles A' and A″, and likewise for Let K_bb and Let K_cc. The lines A-to-K_aa, B-to-Let K_bb, C-to-Let K_cc concur in X₃₅₉₆. For more, see

Boris Odehnal, “Some Triangle Centers Associated with the Circles Tangent to the Excircles,” http://forumgeom.fau.edu/FG2010volume10/FG201006.pdf

-2 a^4 (b+c)-(b-c)^2 (b+c)^3+a (b^2-c^2)^2+a^2 (b+c) (b^2+c^2)-a^3 (3 b^2+4 b c+3 c^2)

: (b-c) c^2 (b+c)^2+a (b-c) c (b+c) (2 b+c)+a^2 (b+c) (2 b^2+c^2)+a^3 (2 b^2+2 b c+c^2)

: (a-b) b^2 (a+b)^2+(a-b) b (a+b) (2 a+b) c+(a+b) (2 a^2+b^2) c^2+(2 a^2+2 a b+b^2) c^3