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X(1), X(2), X(6), X(75), X(239), X(291), X(366), X(518), X(673), X(1575), X(2319), X(2669), X(3212), X(3226), X(7061), X(8301), X(9278) isogonal conjugates of all the points of K155 tX726 = isotomic conjugate of X(726) = X(3226) harmonic associates of X(366) infinite points of the Steiner ellipse A1 = - bc : b^2 : c^2 on AK, B1 and C1 similarly A2 = - a : c : b on AX(75), B2 and C2 similarly |
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K323 is the isogonal transform of K155 = EAC2 = equal-areas (second) cevian cubic, the isotomic transform of K766 = pK(X75, X350) and the G-Hirst transform of K770. It is a weak cubic anharmonically equivalent to K131 as in Table 67. It meets the line at infinity at the same points as the Steiner ellipse and has only one real asymptote. See a generalization in CL041. K323 is the locus of P whose cevian triangle is perspective (at Q) to the 2nd Sharygin triangle. The locus of Q is K961 = pK(X1914, X8301). Compare with K132. See also K673. *** K323 is the cornerstone of a group of 12 cubics all related between themselves under isogonal, isotomic, G-Hirst conjugations – denoted g, t, h in the following diagram – or a product of these such as e = gtg which is X(32)-isoconjugation. All these cubics are weak cubics and contain a good number of ETC centers. A similar group of strong cubics is obtained when K323 is replaced with K718. See also Table 68. |
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