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X(4), X(147), X(315), X(511), X(1297), X(1670), X(1671), X(1916), X(5002), X(5003), X(32618), X(32619), X(34137), X(34214) Ga, Gb, Gc : vertices of antimedial triangle |
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Geometric properties : |
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K1131 is the anticomplement of K570 and the isogonal transform of K1129. K1131 is the locus of pivots of circular pKs which pass through X(511). The locus of the isopivots is K570. See K337 for instance. K1131 is a circular cubic with singular focus X(6033). The real asymptote is the parallel at the Tarry point X(98) to the Brocard axis. The tangents at Ga, Gb, Gc to K1131 concur at X(14957) hence K1131 is a psK in the antimedial triangle with pseudo-pivot X(2) and pseudo-isopivot X(14957). K1131 is a member of the pencil of circular cubics generated by K305 and K337. This also contains the isogonal transform K1134 of K570 and the decomposed cubic into the line at infinity and the Jerabek hyperbola. All these circular cubics pass through A, B, C, X(4), X(511), X(32618), X(32619) and have their singular focus on the Fermat axis. The pencil is stable under antigonal conjugation and two homologous cubics have their singular foci symmetric about X(115). Note that K337 is its own antigonal image with singular focus X(115). The antigonal image of K1131 is K1134. The antigonal image gigK305 of K305 is another cubic of the pencil, with singular focus X(11632). gigK305 passes through X(i) for these i : {2, 4, 23, 111, 263, 511, 11185, 32618, 32619}. The two members with foci X(13), X(14) are the focal cubics K1132a, K1132b, each being the antigonal image of the other. |