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X(1), X(41), X(56), X(208), X(603), X(16944), X(16945), X(17103), X(17104), X(17105), X(17106), X(17107), X(17108), X(17109), X(17110) Q1 = a^3 (a+b2 c) (a2 b+c) (a^2b^2+b cc^2) : : = X(16944) Q2 = a^3 (a+b3 c) (a+bc) (a3 b+c) (ab+c) : : = X(16945) vertices of the cevian triangle of X(57) points of pK(X604, X56) on (O), see below 

Geometric properties : 

A variable line (L) passing through the incenter X(1) of ABC meets the Thomson cubic K002 again at M1, M2 whose barycentric product is S. When (L) rotates around X(1), the locus of S is K967, a nodal cubic with node X(56). The pseudoisopivot is X(31) hence its polar conic (C) is the circumconic passing through X(56) meeting K967 again at Q1 as above. K967 is the barycentric product of : • X(1) and K577 = psK(X56, X7, X2), • X(57) and K259 = psK(X55, X2, X1), • X(6) and K970 = psK(X7, X85, X7), a nodal cubic with node X(7), with asymptotes concurring at the midpoint X(6173) of X(2)X(7), passing through X(7), X(9), X(75), X(77), X(342), • X(7) and psK(X2175, X1, X6), a nodal cubic with node X(6), passing through X(6), X(9), X(48), X(1253), X(1419), X(2191), X(2331). K967 is also the X(31)isoconjugate of K220 = psK(X55, X2, X6), the X(604)isoconjugate of K360 = psK(X56, X7, X1). The isogonal transform of K967 is K971 = psK(X8, X75, X1), a nodal cubic with node X(8), with asymptotes concurring at the midpoint X(3679) of X(2)X(8), passing through X(1), X(8), X(85), X(271), X(318). *** Points on the circumcircle (O) and related cubics A cubic pK(Ω, P) meeting (O) as K967 must have : • its pole Ω on psK(X32 x X1408, X1408, X604) passing through X(604), X(1333), • its pivot P on psK(X1408, X1014, X1) passing through X(1), X(56), X(58). The more simple is probably pK(X604, X56) passing through X(1), X(56), X(266), X(978). Also : • pk1 = pK(Ω1 = a^3 (a^2+a b+a c2 b c) : : , X1), Ω1 = X(16946) now in ETC, • pk2 = pK(Ω2 = a^4 (a+b) (a+bc) (a+c) (ab+c) : : , X58), Ω2 = X(16947) now in ETC, • pk3 = pK(X1333, P3 = a (a+b) (3 abc) (a+c) : : ), P3 = X(16948) now in ETC, with rather simple poles or pivots. *** Generalization A variable line (L) passing through the incenter X(1) of ABC meets the cubic pK(X6, Q) again at M1, M2 whose barycentric product is S. Denote by M the QCeva conjugate of X(1), a point on the pK. When (L) rotates around X(1), the locus of S is the nodal psK with • pseudopole Ω = X(560) ÷ M (barycentric quotient), • pseudopivot P = Q x M* (barycentric product of Q and isogonal conjugate M* of M), • pseudoisopivot X(1) x Q*, • node X(1) x M*. The isogonal transform of this psK is a psK+ with • pseudopole Ω1 = M ÷ X(1) = anticomplement of (X(1) ÷ Q), • pseudopivot P1 = Q ÷ X(1), • pseudoisopivot M ÷ Q, • node N = Ω1, • asymptotes concurring at X, the midpoint of X(2), N. Now, if the node N is given, this psK+ is psK(N, tcN, N) corresponding to pK(X6, X(1) x tcN). For example, – with N = X(8) hence tcN = X(75), we obtain psK+(X8, X75, X8) corresponding to pK(X6, X2) = K002 as above. – with N = X(69) hence tcN = X(76), we obtain psK+(X69, X76, X69) corresponding to pK(X6, X75) = K968.
