X₂₉₆₈

2nd MACBEATH POINT

X₂₉₆₈ = complement of X₁₈₉₇

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

X₂₉₆₈ (2nd MACBEATH POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 1897}

• Related Centers: X₂ (CENTROID), X₁₈₉₇ (ZOSMA TRANSFORM OF X₁₀₅₄)

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

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

Search Value (6, 9, 13): \(4.664644759214\)

▸ Alternate Coordinates1 form

• Trilinears: directed distance of A to line X₁₀₈X₁₀₉ : :

▸ Incident Central Lines18 lines

{2, 1897}, {3, 8}, {4, 280}, {5, 318}, {11, 123}, {78, 1062}, {116, 122}, {124, 522}, {189, 972}, {200, 1040}, {216, 594}, {239, 441}, {253, 1119}, {345, 1260}, {521, 1364}, {525, 2632}, {867, 2969}, {1465, 1861}

▸ Functional Relations9 relations

• X₂₉₆₈ = reflection of X₃₈₅₅₄ in X₃

• X₂₉₆₈ = complement of X₁₈₉₇

• X₂₉₆₈ = anticomplement of X₁₅₂₅₂

• X₂₉₆₈ = X_i-Ceva conjugate of X_j for these (i,j): (8, 521), (75, 525), (253, 514), (280, 522), (2370, 900)

• X₂₉₆₈ = crosspoint of X₁₈₉ and X₆₉₃

• X₂₉₆₈ = crosssum of X_i and X_j for these (i,j): (184, 1415), (198, 692)

• X₂₉₆₈ = polar conjugate of X₂₃₉₈₄

• X₂₉₆₈ = perspector of circumconic centered at X₃₂₃₉

• X₂₉₆₈ = X₂-Ceva conjugate of X₃₂₃₉

Notes

X₂₉₆₈ is the center of circumconic (hyperbola {{A, B, C,X₈,X₂₈₀}}) that is the locus of trilinear poles of lines passing through X₃₂₃₉. This hyperbola is the isotomic conjugate of the Soddy line. (Randy Hutson, October 15, 2018)