X₂₉₇

X₂-HIRST INVERSE OF X₄

X₂₉₇ is the X₂-hirst inverse of X₄.

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

X₂₉₇ (X₂-HIRST INVERSE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {340, 648}, {2, 3} (Euler line)

• Related Centers: X₃₄₀ (ISOTOMIC CONJUGATE OF X₂₆₅), X₆₄₈ (TRILINEAR POLE OF EULER LINE), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(\csc 2A \cos(A + \omega ) : \csc 2B \cos(B + \omega ) : \csc 2C \cos(C + \omega )\)

Barycentrics: \(\sec A \cos(A + \omega ) : \sec B \cos(B + \omega ) : \sec C \cos(C + \omega )\)

Search Value (6, 9, 13): \(2.101911261076\)

▸ Alternate Coordinates3 forms

• Barycentrics: \(\tan A - \cot \omega : :\)

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

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

▸ Linear Combinations1 combination

• X₂₉₇ = X₃₄₀ + \(2\) X₁₉₉₀

▸ Incident Central Lines37 lines

{2, 3}, {6, 317}, {53, 141}, {69, 393}, {76, 343}, {83, 275}, {92, 257}, {107, 2710}, {112, 2857}, {193, 1249}, {230, 2966}, {232, 325}, {239, 5081}, {249, 316}, {273, 3662}, {286, 1865}, {287, 685}, {290, 1987}, {315, 394}, {318, 3661}, {324, 1235}, {340, 524}, {459, 2996}, {525, 850}, {530, 6111}, {531, 6110}, {623, 6117}, {624, 6116}, {626, 3199}, {1515, 1533}, {1654, 2322}, {1785, 3912}, {1915, 1970}, {3087, 3618}, {3978, 6331}, {3981, 5254}, {5032, 5702}

▸ Functional Relations27 relations

• X₂₉₇ = midpoint of X₃₄₀ and X₆₄₈

• X₂₉₇ = reflection of X_i in X_j for these (i,j): (401, 441), (648, 1990), (2966, 230)

• X₂₉₇ = isogonal conjugate of X₂₄₈

• X₂₉₇ = isotomic conjugate of X₂₈₇

• X₂₉₇ = inverse-in-orthocentroidal-circle of X₄₅₈

• X₂₉₇ = complement of X₄₀₁

• X₂₉₇ = anticomplement of X₄₄₁

• X₂₉₇ = X_i-Ceva conjugate of X_j for these (i,j): (264, 2967), (6330, 2)

• X₂₉₇ = cevapoint of X₂₃₂ and X₅₁₁

• X₂₉₇ = X_i-cross conjugate of X_j for these (i,j): (511, 325), (2967, 264), (3569, 4230)

• X₂₉₇ = crossdifference of every pair of points on line X₁₈₄X₆₄₇

• X₂₉₇ = X_i-Hirst inverse of X_j for (i,j) = (2,4), (193,1249)

• X₂₉₇ = X_i-complementary conjugate of X_j for these (i,j): (1953, 129), (1956, 141), (1972, 2887), (1987, 10)

• X₂₉₇ = X_i-anticomplementary conjugate of X_j for these (i,j): (1297, 4329), (6330, 6327}

• X₂₉₇ = perspector of conic {A, B, C,PU(45)}

• X₂₉₇ = intersection of trilinear polars of P(45) and U(45)

• X₂₉₇ = trilinear pole of line X₁₁₄X₁₃₂ (the polar of X₉₈ wrt polar circle)

• X₂₉₇ = pole wrt polar circle of trilinear polar of X₉₈ (line X₆X₅₂₃)

• X₂₉₇ = X₄₈-isoconjugate (polar conjugate) of X₉₈

• X₂₉₇ = inverse-in-polar-circle of X₁₃₁₆

• X₂₉₇ = inverse-in-Steiner-circumellipse of X₄

• X₂₉₇ = {X₂₄₇₉,X₂₄₈₀}-harmonic conjugate of X₄

• X₂₉₇ = inverse-in-Steiner-inellipse of X₅

• X₂₉₇ = {X₂₄₅₄,X₂₄₅₅}-harmonic conjugate of X₅

• X₂₉₇ = midpoint of polar conjugates of PU(4)

• X₂₉₇ = X_i-isoconjugate of X_j for these (i,j): {1, 248}, {3, 1910}, {6, 293}, {31, 287}, {32, 336}, {48, 98}, {63, 1976}, {163, 879}, {184, 1821}, {656, 2715}, {662, 878}, {685, 822}, {810, 2966}, {1176, 3404}, {1973, 6394}, {2395, 4575}, {2422, 4592}

• X₂₉₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 4, 458), (2, 401, 441), (4, 419, 460), (4, 420, 419), (4, 5117, 427), (53, 141, 264), (237, 868, 2450), (237, 2450, 1513), (419, 420, 468), (460, 468, 419), (470, 471, 468), (472, 473, 428), (868, 5112, 1513), (1585, 1586, 25), (2450, 5112, 237), (2454, 2455, 5), (2479, 2480, 4)

Notes

As a point on the Euler line, X₂₉₇ has Shinagawa coefficients \((EF + F^{2}, - S^{2})\).