X₂₉₇₀

4th MACBEATH POINT

X₂₉₇₀ = X₈₄₇-Ceva conjugate of X₅₂₃

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

X₂₉₇₀ (4th MACBEATH POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 94}, {25, 98}

• Related Centers: X₄ (ORTHOCENTER), X₉₄ (ISOGONAL CONJUGATE OF X₅₀), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₉₈ (TARRY POINT)

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

Barycentrics: \((\tan^{2} A) (\tan B - \tan C)^{2} ((\tan A) (\tan A + \tan B + \tan C) + \tan B \tan C)) : :\)

Search Value (6, 9, 13): \(3.126706411984\)

▸ Alternate Coordinates1 form

• Barycentrics: \(b^{2} c^{2} (b^{2} - c^{2})^{2}/(a^{2} - b^{2} - c^{2}) : :\)

▸ Incident Central Lines8 lines

{2, 2974}, {3, 847}, {4, 94}, {25, 98}, {115, 135}, {125, 136}, {324, 427}, {339, 868}

▸ Functional Relations9 relations

• X₂₉₇₀ = reflection of X₃₄₃₃₃ in X₅

• X₂₉₇₀ = MacBeath-inconic-antipode of X₃₄₃₃₃

• X₂₉₇₀ = X₈₄₇-Ceva conjugate of X₅₂₃

• X₂₉₇₀ = X₁₁₅-cross conjugate of X₃₃₈

• X₂₉₇₀ = crosssum of X₅₇₁ and X₁₅₇₆

• X₂₉₇₀ = pole wrt polar circle of trilinear polar of X₂₄₉ (line X₁₁₀X₃₅₁)

• X₂₉₇₀ = polar conjugate of X₂₄₉

• X₂₉₇₀ = X₃-isoconjugate of X₁₁₀₁

• X₂₉₇₀ = {X₃₉₂₄₀,X₃₉₂₄₁}-harmonic conjugate of X₃₅₂₃₅