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For any pivot P, the cubic K = pK(O x P, P) has isopivot O = X(3) hence its tangents at A, B, C, P concur at O. The isogonal transform K* of K is a member of CL019 with pivot H. If P = O, the cubic splits into the cevian lines of O. K passes through O, P, the P-Ceva conjugate P/O of O (which is the tangential of O and the tertiary pivot of K) and the crossconjugate O©P of O and P. K meets the line at infinity (L∞) and the circumcircle (O) at the same points as two isogonal pKs with respective pivots P1, P2. We have P1 = a(ctP ÷ H) and P2 = agP. See CL073 for a generalization. pK(O x P, P) passes through a given point M ≠ O if and only if P lies on the circum-conic C(M) with perspector the barycentric quotient of M^2 and the trilinear pole of the line OM. In this case, the cubic must contain O©M and these cubics form a pencil. K is circular if and only if P2 = agP is on (L∞) hence P is on (O). In this case, K is an inversible cubic and its singular focus lies on the circumcircle of the tangential triangle. K is equilateral (i.e. is a pK60) if and only if P1 = O hence P = X(15318). K is a pK+ if and only if P lies on a circum-cubic passing through X(3), X(5), X(20), PM1 = {3,6}/\{217,1154}, PM2 = {3,3164}/\{5,324}. This cubic meets (L∞) at the same points as pK(X6, X34148) and (O) at the same points as pK(X6, PM3) where PM3 = {20,2979}/\{30,49}. These points PM1, PM2, PM3 are now X(41480), X(41481), X(41482) in ETC. The locus of the intersection of the asymptotes is then an equilateral cubic passing through X(2), X(3), X(154), X(1075), X(14059), the cevians of O, the infinite points of K003. *** The following table (contributed by Peter Moses) shows a large selection of these cubics. Notes • When P lies on the Kiepert hyperbola, the pole Ω = O x P of K lies on the Jerabek hyperbola. These cubics form a pencil and pass through X(485), X(486). The third point of K on the line through X(485), X(486) is the P-Ceva conjugate P/O of O. See green cells in the table. • When P lies on the Feuerbach hyperbola, the pole Ω = O x P of K lies on the circum-hyperbola with perspector X(1946) passing through X(6), X(48), etc. These cubics form another pencil and pass through X(1), X(90). The third point of K on the line through X(1), X(90) is the P-Ceva conjugate P/O of O. See blue cells in the table. • When P lies on the circum-hyperbola with perspector X(525), K passes through X(2) and X(69) and Ω lies on the circum-hyperbola with perspector X(520). this gives another pencil corresponding to the purple cells in the table. • The grey cells show the cubics passing through X(6) and X(8770) for P on the circum-conic with perspector X(512). The pink cells show the cubics passing through X(4) and X(254) for P on the circum-conic with perspector X(2501). • The red cells show cubics that belong to at least two of these pencils. |
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