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Let P = p : q : r be a fixed point. M = x : y : z is a variable point and agM is the anticomplement of its isogonal conjugate. The locus of M such that P, M, agM are collinear is the cubic K(P) = psK(gcP, tcP, P) = spK(cP, ccP). Its equation is : ∑ a^2 y z (q xr xp yr y+p z+q z) = 0. Properties K(P) passes through : • A, B, C where the tangents are the symmedians. • the orthocenter H of ABC. • P and gcP. • the infinite points of pK(X6, cP). The six common finite points of the two cubics lie on the isogonal transform of the line GP. • the points of pK(X6, P) on the circumcircle (O), namely A, B, C and three other points Q1, Q2, Q3 not always all real. The three other common points of the two cubics are P and two points on the line GP. Note that the tangents at Q1, Q2, Q3 to K(P) concur (at X) for all P hence K(P) is also a psK with respect to the triangle Q1Q2Q3. When Plies on the anticomplements of the cubic K652 or the deltoid H3, this point X lies on K(P) which becomes a pK in Q1Q2Q3 This occurs in particular when P = O hence K(P) = K361 with X = O, also when P = X(3146) hence K(P) = K841 with X = X(64). • the foci of the inconic with center ccP, perspector tcP. Indeed, K(P) and its isogonal transform gK(P) = spK(P, ccP) are two members of a pencil which contains pK(X6, ccP). • a given point Q ≠ H if and only if P lies on the line through Q and agQ. For instance, K(P) passes through O when P is on the Euler line. K(P) is globally invariant under two transformations 𝛗_{1} and 𝛗_{2} which are inverse of one another : for any point M on K(P), M_{1} = 𝛗_{1} (M) and M_{2} = 𝛗_{2} (M) are also two points on K(P). M_{1} is the tcPCeva conjugate of the gcPisoconjugate of M and M_{2} is the gcPisoconjugate of the tcPCeva conjugate of M. With M = x : y : z, this gives : M_{1} = (a^2 (c^2 x yb^2 x z+a^2 y z)) / ((q+r) x) : : and M_{2} = a^2 / ((q+r) x (q x+r xp yr yp zq z)) : : . *** Special cubics • K(P) is a pK if and only if it is a K0 i.e. if and only if P lies on the line GK. In this case it must contain its isopivot K and also G. These pKs belong to a same pencil generated by K002 (P = G) and K644 (P = K). See table below for other examples and also CL043. • K(P) is circular if and only if P lies on the line at infinity. The singular focus lies on the circle C(O, 2R) passing through X(399), the focus of K568 obtained when P = X(523). • K(P) is equilateral if and only if P = H giving the stelloid K028. • K(P) is a K+ (with concurring asymptotes) if and only if P lies on a cicumcubic of the antimedial triangle passing through H (giving K028), X(20) (giving K443) and X(2889). The complement of this cubic is a circumcubic meeting the line at infinity like pK(X6, X140) and (O) like K003 i.e. at the vertices of the CircumNormal triangle. • K(P) is a nodal (unicursal) cubic if and only if P lies ona complicated sextic passing through H (giving K028), X(145) (giving K360). The following table gives a list of catalogued cubics K(P). 




