X₂₉₆₆

BARYCENTRIC PRODUCT X₉₈*X₉₉

X₂₉₆₆ = midpoint of X₃₈₅ and X₄₀₁

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

X₂₉₆₆ (BARYCENTRIC PRODUCT X₉₈*X₉₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {385, 401}

• Related Centers: X₃₈₅ (-HIRST INVERSE OF X₆), X₄₀₁ (BAILEY POINT)

Trilinears: \([\sec(A + \omega )]/(b^{2} - c^{2}) : :\)

Search Value (6, 9, 13): \(-1.718106536759\)

X₂₉₆₆ lies on the Steiner circumellipse, the hyperbola {{A, B, C,X₂,X₄₇₆}}, the conic {{A, B, C,X₁₁₂,X₂₄₈}}.

▸ Incident Central Lines12 lines

{30, 98}, {99, 249}, {230, 297}, {248, 290}, {250, 523}, {287, 524}, {325, 441}, {448, 2481}, {668, 906}, {879, 4266}, {892, 2395}, {2421, 2422}

▸ Functional Relations12 relations

• X₂₉₆₆ = midpoint of X₃₈₅ and X₄₀₁

• X₂₉₆₆ = reflection of X_i and X_j for these (i,j): (297, 230), (325, 441)

• X₂₉₆₆ = isogonal conjugate of X₃₅₆₉

• X₂₉₆₆ = isotomic conjugate of X₂₇₉₉

• X₂₉₆₆ = anticomplement of X₃₅₀₈₈

• X₂₉₆₆ = polar conjugate of X₁₆₂₃₀

• X₂₉₆₆ = complement of X₃₉₃₅₉

• X₂₉₆₆ = trilinear pole of line X₂X₉₈

• X₂₉₆₆ = antipode of X₂ in hyperbola {{A, B, C,X₂,X₄₇₆}}

• X₂₉₆₆ = cevapoint of X_i and X_j for these (i,j): (2, 2799), (98, 2395), (112, 2409), (230, 523), (248, 879), (441, 525), (511, 647)

• X₂₉₆₆ = X_i-cross conjugate of X_j for these (i,j): (879, 290), (2395, 98), (2715, 685), (2799, 2)

• X₂₉₆₆ = barycentric product of Steiner circumellipse intercepts of line X₂X₉₈

Notes

Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X₂₉₆₆ is the point of intersection of the lines P(2)U(40) and P(40)U(2). (Peter Moses, July 1, 2009). See also X₃₅₆₈.