X₃₆₆₀

MIDPOINT OF {X₃₅₁₃, X₃₅₁₄}

X₃₆₆₀ is the midpoint of {X₃₅₁₃, X₃₅₁₄}.

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

X₃₆₆₀ (MIDPOINT OF {X₃₅₁₃, X₃₅₁₄}) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Incircle (incircle) — Inscribed circle tangent to sidelines at T_a, T_b, T_c; centered at X₁ INCENTER

• Central Lines: {1, 57}

• Related Centers: X₁ (INCENTER), X₅₇ (ISOGONAL CONJUGATE OF X₉)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = r^{2}(r + 4R) + (r - 2R)s_{B}s_{C}\)

Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(a - b + c)(a + b - c)(b^{3} + c^{3} + a^{2}b + a^{2}c - 2ab^{2} - 2ac^{2} + 2abc - b^{2}c - bc^{2})\)

Search Value (6, 9, 13): \(-0.199435275575\)

▸ Linear Combinations1 combination

• X₃₆₆₀ = \((r^{2} + 2rR -2R^{2})\) X₁ - \(r(r - 2R)\) X₃

▸ Incident Central Lines17 lines

{1, 3}, {11, 971}, {105, 2720}, {108, 840}, {109, 1279}, {222, 614}, {244, 1458}, {513, 676}, {518, 3035}, {1071, 3086}, {1357, 3322}, {1360, 3025}, {1404, 2246}, {1421, 1456}, {1426, 1878}, {1477, 2222}, {1788, 3555}

▸ Functional Relations8 relations

• X₃₆₆₀ = isogonal conjugate of X₃₄₈₉₄

• X₃₆₆₀ = inverse-in-incircle of X₅₇

• X₃₆₆₀ = X₄₆₈-of-intouch-triangle

• X₃₆₆₀ = inverse-in-{circumcircle, nine-point circle}-inverter of X₅₆

• X₃₆₆₀ = radical trace of the incircle and circumcircle; see X₃₅₁₃

• X₃₆₆₀ = radical trace of circumcircle and Moses-Longuet-Higgins circle

• X₃₆₆₀ = radical trace of incircle and Moses-Longuet-Higgins circle

• X₃₆₆₀ = inverse-in-circumcircle of X₁₆₁₇

Notes

Let CA be the circumcircle analog of the Conway circle; that is, the circle with center X₃ and radius (\(R^{2}\) + \(s^{2})\)^1/2. Then X₃₆₆₀ is the radical trace of CA and the Conway circle. (Randy Hutson, October 13, 2015)