X₂₉₆₃

BARYCENTRIC PRODUCT X₁₇*X₁₈

,p> X₂₉₆₃ lies on these lines: 2,1225 5,2965 6,17 25,2934 37,2962 50,252 53,1487 93,393 111,930 141,2987 216,1989 230,251 566,2165 588,590 589,615 1879,2937

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

X₂₉₆₃ (BARYCENTRIC PRODUCT X₁₇*X₁₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Related Centers: X₅₉₀ (ISOGONAL CONJUGATE OF X₅₈₈), X₆₁₅ (ISOGONAL CONJUGATE OF X₅₈₉)

Trilinears: \(\sin A \csc(A + \pi /6) \csc(A - \pi /6) : :\)

Barycentrics: \(1/(3 - \cot^{2} A) : :\)

Search Value (6, 9, 13): \(-5.528814688072\)

▸ Alternate Coordinates1 form

• Barycentrics: \(1/(a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2} - b^{2} c^{2}) : :\)

▸ Functional Relations11 relations

• X₂₉₆₃ = isogonal conjugate of X₁₉₉₄

• X₂₉₆₃ = cevapoint of X₅₉₀ and X₆₁₅

• X₂₉₆₃ = X₅₇₀-cross conjugate of X₆

• X₂₉₆₃ = crosssum of X_i and X_j for these (i,j): (6, 195), (49, 2965)

• X₂₉₆₃ = isotomic conjugate of X₇₇₆₉

• X₂₉₆₃ = {X₁₇,X₁₈}-harmonic conjugate of X₃₅₁₉

• X₂₉₆₃ = X₂-isoconjugate of X₂₉₆₄

• X₂₉₆₃ = X₉₂-isoconjugate of X₄₉

• X₂₉₆₃ = polar conjugate of X₃₂₀₀₂

• X₂₉₆₃ = perspector of ABC and unary cofactor triangle of Kosnita triangle

• X₂₉₆₃ = barycentric product of nine-point centers of 1st and 2nd Ehrmann inscribed triangles

Notes

Let Q be the pedal curve (a limaçon of Pascal) of the circle with center X₇₇₂₈ and radius |OH|, with respect to X₂₆₅. Let A'B'C' be the triangle formed by the tangents at A, B, C to Q. Then the triangles ABC and A'B'C' are perspective, and their perspector is X₂₉₆₃. Moreover, X₂₁₉₇₅ is the only finite fixed point of the affine transformation that maps a triangle ABC onto A'B'C'. See X₂₁₉₇₅ (X(21975).png). (Angel Montesdeoca, September 10, 2019)

In the plane of a triangle ABC, let

• N= Nine point circle,

• N'_a, A' = reflection of N, A in BC, define N'_b, B' and N'_c, C' cyclically

• P_a = polar of A' with respect to N'_a, and define P_b and P_c cyclically;

• A″ = P_c∩P_b, and define B″ and C″ cyclically.

The triangle A″B″C″ is perspective to ABC, and the perspector is X₂₉₆₃. (Dasari Naga Vijay Krishna, June 8, 2021)

Generalization of Vijay’s construction of X₂₉₆₃. Let C(U) = the circle with powers u, v, w with respect to the vertices A, B, C, respectively;

• C'(U) = reflection of C(U) in side BC;

• A' = reflection of A in in side BC;

• P_a = polar of A in C'(U), and define P_b and P_c cyclically;

• A″ = P_c∩P_b, and define B″ and C″ cyclically.

• The triangle A″B″C″ is perspective to ABC, and the perspector is

• a^2 / (a^2*(a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2 - c^4) + a^2*(b^2 + c^2 - u)*u - (a^4 + a^2*b^2 - b^4 + 2*b^2*c^2 - c^4)*v + (a^2 + b^2 - c^2)*v^2 - (a^4 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*w + (a^2 - b^2 + c^2)*w^2 - a^2*(u*v + u*w - v*w)) : : (Peter Moses, June 11, 2021)