X₁₅₉

X₉-OF-TANGENTIAL-TRIANGLE

X₁₅₉ = X₉-of-tangential triangle if ABC is acute

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

X₁₅₉ (X₉-OF-TANGENTIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Tangential Triangle △Tan_aTan_bTan_c — vertices Tan_a, Tan_b, Tan_c (Bounded by tangents to the circumcircle at A, B, C)

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Concurrence Lines: Tan_a–X₁₅₉, Tan_b–X₁₅₉, Tan_c–X₁₅₉ concurring at X₁₅₉

• Related Centers: X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[(a^{2} + b^{2} + c^{2})\sin 2A + (c^{2} - b^{2} - a^{2})\sin 2B + (b^{2} - c^{2} - a^{2})\sin 2C]\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(-2.745390492160\)

▸ Incident Central Lines6 lines

{3, 66}, {6, 25}, {22, 69}, {23, 193}, {155, 511}, {197, 200}

▸ Functional Relations12 relations

• X₁₅₉ = X₉-of-tangential triangle if ABC is acute

• X₁₅₉ = reflection of X_i in X_j for these (i,j): (6, 206), (66, 141)

• X₁₅₉ = isogonal conjugate of X₁₃₅₇₅

• X₁₅₉ = crosspoint of X₁₁₀ and X₁₅₃₈₈

• X₁₅₉ = crossdifference of every pair of points on line X₅₂₅X₂₄₈₅

• X₁₅₉ = trilinear product X_i*X_j for these {i,j}: {31, 1370}, {38, 8793}, {63, 3162}

• X₁₅₉ = complement of X₃₆₈₅₁

• X₁₅₉ = X_i-Ceva conjugate of X_j for these (i,j): (22, 3), (69, 6)

• X₁₅₉ = crosssum of X_i and X_j for these {i,j}: {2, 7500}, {127, 523}

• X₁₅₉ = tangential-isogonal conjugate of X₂₅

• X₁₅₉ = tangential-isotomic conjugate of X₃₂₄₄₅

• X₁₅₉ = intouch-to-ABC functional image of X₉