X₂₀₇₈

INVERSE-IN-CIRCUMCIRCLE OF X₅₇

X₂₀₇₈ = inverse-in-circumcircle of X₅₇

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

X₂₀₇₈ (INVERSE-IN-CIRCUMCIRCLE OF X₅₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

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

• Central Lines: {3, 57}

• Related Centers: X₃ (CIRCUMCENTER), X₅₇ (ISOGONAL CONJUGATE OF X₉)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = r - 2R + 2(r + R) \cos A\)

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

Search Value (6, 9, 13): \(-0.660678156305\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \cos B + \cos C - 3 + (1 + 2 \cos A + 2 \cos B + 2 \cos C) \cos A\)

• Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a(a^{2} + b^{2} + c^{2} - 2ab - 2ac + bc)/(b+c-a)\) (M. Iliev, 5/13/07)

▸ Incident Central Lines14 lines

{1, 3}, {31, 2003}, {59, 672}, {73, 595}, {105, 2006}, {109, 840}, {226, 1005}, {388, 535}, {581, 1497}, {901, 1477}, {1174, 1202}, {1279, 1421}, {1283, 1284}, {1308, 1323}

▸ Functional Relations5 relations

• X₂₀₇₈ = isogonal conjugate of X₃₂₅₄

• X₂₀₇₈ = inverse-in-circumcircle of X₅₇

• X₂₀₇₈ = crosssum of X₁₄₂ and X₅₂₇

• X₂₀₇₈ = inverse-in-{circumcircle, incircle}-inverter of X₆₅

• X₂₀₇₈ = {X₃₅₁₃,X₃₅₁₄}-harmonic conjugate of X₃₆