X₂₆₄

ISOTOMIC CONJUGATE OF CIRCUMCENTER

X₂₆₄ is the isotomic conjugate of circumcenter.

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

X₂₆₄ (ISOTOMIC CONJUGATE OF CIRCUMCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3164, 216}

• Related Centers: X₃₁₆₄ (-CEVA CONJUGATE OF X₂), X₂₁₆ (-CEVA CONJUGATE OF X₅₁)

Trilinears: \(\csc A \csc 2A : \csc B \csc 2B : \csc C \csc 2C\)

Barycentrics: \(\csc 2A : \csc 2B : \csc 2C\)

Search Value (6, 9, 13): \(8.859306901023\)

▸ Alternate Coordinates6 forms

• Trilinears: \(\sec A \csc^{2} A : \sec B \csc^{2} B : \sec C \csc^{2} C\)

• Trilinears: \(\tan A \csc(A - \omega ) : \tan B \csc(B - \omega ) : \tan C \csc(C - \omega )\)

• Trilinears: \(\sec A + \cot A \csc A : :\)

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

• Barycentrics: \(\tan A + \cot A : :\)

• Barycentrics: \(\tan A - \cot B - \cot C + \cot \omega : :\)

X₂₆₄ lies on these cubics: K045, K146, K183, K184, K208, K257, K276, K342a, K342b, K494, K504, K647, K674, K675, K677, K718

▸ Incident Central Lines94 lines

{2, 216}, {3, 95}, {4, 69}, {5, 1093}, {6, 287}, {9, 1948}, {22, 1629}, {24, 1078}, {25, 183}, {33, 350}, {34, 1909}, {53, 141}, {57, 1947}, {75, 225}, {85, 309}, {92, 306}, {93, 1273}, {98, 3425}, {99, 378}, {107, 1995}, {112, 2367}, {157, 1485}, {186, 7771}, {193, 3087}, {250, 1316}, {253, 3091}, {254, 3541}, {262, 2967}, {274, 475}, {275, 1993}, {281, 344}, {298, 472}, {299, 473}, {300, 302}, {301, 303}, {305, 325}, {310, 4196}, {319, 5081}, {320, 7282}, {328, 6344}, {339, 381}, {379, 823}, {384, 1968}, {401, 577}, {419, 1974}, {450, 5651}, {491, 1585}, {492, 1586}, {524, 6748}, {623, 6116}, {624, 6117}, {801, 2063}, {811, 5136}, {847, 1594}, {850, 7703}, {1007, 6340}, {1043, 7513}, {1105, 1593}, {1217, 3088}, {1225, 7809}, {1238, 7796}, {1249, 3618}, {1309, 2861}, {1441, 2476}, {1595, 3933}, {1726, 7094}, {1785, 4357}, {1896, 2478}, {1897, 4360}, {1969, 3262}, {1990, 3589}, {2207, 7770}, {2419, 3267}, {2453, 3447}, {2897, 6840}, {2970, 5094}, {3148, 6394}, {3168, 5943}, {3199, 3934}, {3520, 7782}, {3575, 7750}, {3629, 6749}, {3785, 7487}, {5064, 7788}, {5117, 6374}, {5523, 7790}, {6103, 7806}, {6240, 7802}, {6524, 7392}, {6525, 7398}, {6756, 7767}, {7378, 8024}, {7507, 7773}, {7576, 7811}

▸ Functional Relations23 relations

• X₂₆₄ = reflection of X₃₁₆₄ in X₂₁₆

• X₂₆₄ = isogonal conjugate of X₁₈₄

• X₂₆₄ = isotomic conjugate of X₃

• X₂₆₄ = complement of X₃₁₆₄

• X₂₆₄ = anticomplement of X₂₁₆

• X₂₆₄ = X₂₆₄ = X_i-Ceva conjugate of X_j for these (i,j): (276, 2), (1969, 7017), (6528, 850)

• X₂₆₄ = cevapoint of X_i and X_j for these (i,j): (2, 4), (5, 324), (6, 157), (92, 318), (273, 342), (338, 523), (491, 492)

• X₂₆₄ = X_i-cross conjugate of X_j for these (i,)}: (2, 76), (3, 5392), (4, 2052), (5, 2), (30, 94), (92, 331), (235, 459), (318, 7017), (339, 850), (427, 4), (442, 321), (523, 648), (850, 6528), (858, 671), (1312, 2593), (1313, 2592), (1368, 2996), (1441, 75), (1591, 5490), (1592, 5491), (1594, 275), (2072, 2986), (2450, 98), (2967, 297), (2968, 4391), (2971, 2501), (2972, 525), (3007, 903), (3134, 2394), (3136, 10), (3141, 4049), (3142, 226), (3143, 5466), (5133, 83), (5169, 598), (6530, 6330), (6563, 99)

• X₂₆₄ = X₁₉₈₈-complementary conjugate of X₁₀

• X₂₆₄ = X_i-anticomplementary conjugate of X_j for these (i,j): (54, 6360), (92, 2888), (95, 4329), (275, 8), (276, 6327), (933, 4560), (2148, 3164), (2167, 20), (2190, 2)

• X₂₆₄ = antipode of X₁₉₇₂ in hyperbola {}A, B, C,X₂,X₆₉}}

• X₂₆₄ = pole of Lemoine axis wrt polar circle

• X₂₆₄ = X₄₈-isoconjugate (polar conjugate) of X₆

• X₂₆₄ = polar-circle inverse of X₅₁₆₇

• X₂₆₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 324, 2052), (2, 3164, 216), (4, 69, 317), (4, 1235, 76), (4, 3186, 1843), (53, 141, 297), (69, 311, 76), (69, 317, 340), (273, 318, 75), (311, 3260, 69)

• X₂₆₄ = Brianchon point (perspector) of the MacBeath inconic

• X₂₆₄ = X_i-isoconjugate of X_j for these {i,j}: {1, 184}, {3, 31}, {6, 48}, {19, 577}, {25, 255}, {28, 4055}, {32, 63}, {33, 7335}, {34, 6056}, {41, 222}, {42, 1437}, {47, 2351}, {51, 2169}, {55, 603}, {56, 212}, {58, 228}, {69, 560}, {71, 1333}, {72, 2206}, {73, 2194}, {77, 2175}, {78, 1397}, {81, 2200}, {97, 2179}, {109, 1946}, {110, 810}, {112, 822}, {163, 647}, {172, 7116}, {213, 1790}, {216, 2148}, {217, 2167}, {219, 604}, {220, 7099}, {221, 2188}, {237, 293}, {248, 1755}, {251, 4020}, {268, 2199}, {283, 1402}, {284, 1409}, {295, 2210}, {304, 1501}, {305, 1917}, {326, 1974}, {393, 4100}, {394, 1973}, {418, 2190}, {512, 4575}, {563, 2165}, {571, 1820}, {607, 7125}, {608, 2289}, {649, 906}, {652, 1415}, {656, 1576}, {662, 3049}, {667, 1331}, {669, 4592}, {692, 1459}, {798, 4558}, {849, 3690}, {895, 922}, {904, 3955}, {923, 3292}, {1092, 1096}, {1106, 1260}, {1110, 3937}, {1176, 1964}, {1253, 7053}, {1259, 1395}, {1332, 1919}, {1399, 8606}, {1400, 2193}, {1407, 1802}, {1408, 2318}, {1410, 2328}, {1433, 2187}, {1444, 1918}, {1472, 7085}, {1473, 7084}, {1474, 3990}, {1797, 2251}, {1798, 3725}, {1799, 1923}, {1804, 2212}, {1813, 3063}, {1910, 3289}, {1911, 7193}, {1914, 2196}, {1924, 4563}, {1949, 1951}, {1950, 7107}, {1980, 4561}, {2149, 7117}, {2150, 2197}, {2159, 3284}, {2192, 7114}, {2203, 3682}, {2207, 6507}, {2208, 7078}, {2300, 2359}, {7011, 7118}, {7015, 7122}

• X₂₆₄ = X_i-beth conjugate of X_j for these (i,j): (264, 273), (811, 7), (3596, 322)

• X₂₆₄ = trilinear pole of the line (297,525)

• X₂₆₄ = barycentric product X_i*X_j for these {i,j}: {1, 1969}, {4, 76}, {5, 276}, {7, 7017}, {8, 331}, {19, 561}, {25, 1502}, {27, 313}, {29, 349}, {69, 2052}, {75, 92}, {83, 1235}, {85, 318}, {93, 7769}, {94, 340}, {95, 324}, {107, 3267}, {158, 304}, {273, 312}, {275, 311}, {278, 3596}, {281, 6063}, {286, 321}, {290, 297}, {300, 470}, {301, 471}, {305, 393}, {308, 427}, {310, 1826}, {317, 5392}, {326, 6521}, {327, 458}, {341, 1847}, {523, 6331}, {525, 6528}, {648, 850}, {670, 2501}, {683, 5254}, {693, 6335}, {811, 1577}, {847, 7763}, {1016, 2973}, {1088, 7101}, {1093, 3926}, {1231, 1896}, {1240, 1848}, {1509, 7141}, {1824, 6385}, {1897, 3261}, {1928, 1973}, {1978, 7649}, {2489, 4609}, {2970, 4590}, {3064, 4572}, {3114, 5117}, {3264, 6336}, {6344, 7799}, {6386, 6591}

• X₂₆₄ = trilinear product of PU(20) (see Tables: Bicentric Pairs)

• X₂₆₄ = trilinear product X_i*X_j for these {i,j}: {2, 92}, {4, 75}, {6, 1969}, {7, 318}, {8, 273}, {9, 331}, {10, 286}, {19, 76}, {25, 561}, {27, 321}, {28, 313}, {29, 1441}, {33, 6063}, {34, 3596}, {57, 7017}, {63, 2052}, {69, 158}, {82, 1235}, {85, 281}, {91, 317}, {162, 850}, {225, 314}, {240, 290}, {242, 334}, {253, 1895}, {274, 1826}, {276, 1953}, {278, 312}, {279, 7101}, {280, 342}, {297, 1821}, {304, 393}, {305, 1096}, {307, 1896}, {309, 7952}, {310, 1824}, {311, 2190}, {324, 2167}, {326, 1093}, {336, 6530}, {340, 2166}, {341, 1119}, {346, 1847}, {347, 7020}, {349, 1172}, {394, 6521}, {419, 1934}, {427, 3112}, {514, 6335}, {523, 811}, {525, 823}, {648, 1577}, {653, 4391}, {656, 6528}, {661, 6331}, {668, 7649}, {693, 1897}, {757, 7141}, {765, 2973}, {799, 2501}, {873, 7140}, {1088, 7046}, {1118, 3718}, {1240, 1829}, {1446, 2322}, {1494, 1784}, {1502, 1973}, {1748, 5392}, {1783, 3261}, {1857, 7182}, {1861, 2481}, {1928, 1974}, {1947, 7108}, {1948, 1952}, {1978, 6591}, {2333, 6385}, {2489, 4602}, {2580, 2592}, {2581, 2593}, {2969, 7035}, {2997, 5125}, {3064, 4554}, {3113, 5117}, {3926, 6520}, {4358, 6336}, {5342, 5936}, {7009, 7018}

• X₂₆₄ = barycentric quotient X_i/X_j for these (i,j): (1, 48), (2, 3), (4, 6), (5, 216), (6, 184), (7, 222), (8, 219), (9, 212), (10, 71), (19, 31), (25, 32), (27, 58), (29, 284), (33, 41), (37, 228), (51, 217), (63, 255), (69, 394), (94, 265), (95, 97), (98, 248), (107, 112), (162, 163), (196, 221), (216, 418), (232, 237), (304, 326), (311, 343), (445, 500)

Notes

Five constructions by Randy Hutson, January 29, 2015:

1. Let Sym_aSym_bSym_c be the tangential triangle. Let A″ be the pole, wrt the polar circle, of line Sym_bSym_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₆₄.

2. Let Sym_aSym_bSym_c be the symmedial triangle. Let A″ be the pole, wrt the polar circle, of line Sym_bSym_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₆₄.

3. Let Sym_aSym_bSym_c be the circumsymmedial triangle. Let A″ be the pole, wrt the polar circle, of line Sym_bSym_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₆₄.

4. Let Sym_aSym_bSym_c be the Lucas(t) central triangle (for any t). Let A″ be the pole, wrt the polar circle, of line Sym_bSym_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₆₄.

5. X₂₆₄ is the trilinear pole of the line X₂₉₇X₅₂₅. This line is the isotomic conjugate of the MacBeath circumconic, which is the isogonal conjugate of the orthic axis. The line is also the polar of X₆ wrt the polar circle, and the radical axis of the polar and orthosymmedial circles, and the polar conjugate of the circumcircle)

Let A' be the trilinear product of the vertices of the A-anti-altimedial triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₆₄. (Randy Hutson, November 2, 2017)