X₃₅₆₉

BELTRAMI-PARRY POINT

X₃₅₆₉ = perspector of hyperbola {{A, B, C,X₆,X₂₃₂}}

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

X₃₅₆₉ (BELTRAMI-PARRY POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {74, 111}, {110, 112}

• Related Centers: X₂₉₆₆ (BARYCENTRIC PRODUCT X₉₈*X₉₉), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₁₁₁ (PARRY POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT))

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b^{2} - c^{2})(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2})\)

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

Search Value (6, 9, 13): \(-4.333626684729\)

X₃₅₆₉ lies on the Walsmith rectangular hyperbola.

▸ Incident Central Lines18 lines

{6, 526}, {74, 111}, {110, 112}, {113, 1560}, {115, 125}, {187, 237}, {248, 2422}, {297, 525}, {520, 2451}, {684, 2491}, {688, 2514}, {694, 804}, {879, 1987}, {924, 2485}, {1499, 1513}, {1510, 3050}, {1769, 2294}, {2450, 3566}

▸ Functional Relations22 relations

• X₃₅₆₉ = isogonal conjugate of X₂₉₆₆

• X₃₅₆₉ = reflection of X_i in X_j for these (i,j): (6, 2492), (1640, 1637), (2451, 2489), (3049, 2485), (3288, 647), (32120, 468)

• X₃₅₆₉ = X_i-Ceva conjugate of X_j for these (i,j): (2, 38987), (297, 868), (648, 2967), (694, 3124), (878, 2451), (1987, 3269), (2421, 511), (2715, 6)

• X₃₅₆₉ = complement of anticomplementary conjugate of X₃₉₃₅₉

• X₃₅₆₉ = perspector of hyperbola {{A, B, C,X₆,X₂₃₂}}

• X₃₅₆₉ = crosspoint of X_i and X_j for these (i,j): (6, 2515), (98, 648), (110, 2987)

• X₃₅₆₉ = crosssum of X_i and X_j for these (i,j): (2, 2799), (98, 2395), (112, 2409), (230, 523), (248, 879), (441, 525), (511, 647)

• X₃₅₆₉ = crossdifference of every pair of points on the line X₂X₉₈

• X₃₅₆₉ = Parry-circle-inverse of X₆₄₇

• X₃₅₆₉ = midpoint of Jerabek hyperbola intercepts of Lemoine axis

• X₃₅₆₉ = Parry-isodynamic-circle-inverse of X₁₄₉₅; see X₂

• X₃₅₆₉ = X₈₅₀-of-1st-Brocard-triangle

• X₃₅₆₉ = X₆-of-2nd-Parry-triangle

• X₃₅₆₉ = X₆-of-3rd-Parry-triangle

• X₃₅₆₉ = 1st-Brocard-isotomic conjugate of X₁₃₁₆

• X₃₅₆₉ = intersection of trilinear polar of X₆ and polar wrt polar circle of X₆

• X₃₅₆₉ = centroid of (degenerate) cross-triangle of 2nd and 3rd Parry triangles

• X₃₅₆₉ = center of the (degenerate) perspeconic of the 2nd and 3rd Parry triangles

• X₃₅₆₉ = antipode of X₃₂₁₂₀ in Walsmith rectangular hyperbola

• X₃₅₆₉ = orthocenter of X₆X₇₄X₁₁₀

• X₃₅₆₉ = orthocenter of X₆X₁₁₃X₁₂₅

• X₃₅₆₉ = orthocenter of X₁₂₅X₁₄₉₅X₃₅₈₀

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 X₃₅₆₉ is the point of intersection of the lines P(2)U(2) and P(40)U(40). The name “Beltrami-Parry Point” signifies the fact that X₃₅₆₉ lies on the line X₇₄X₁₁₁; where X₁₁₁ is the Parry point.and X₇₄ is the isogonal conjugate of the Euler infinity point. See also X₃₅₆₈. Contributed by Chris van Tienhoven, June 7, 2009..