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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 x-r x-p y-r 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 𝛗₁ and 𝛗₂ which are inverse of one another : for any point M on K(P), M₁ = 𝛗₁ (M) and M₂ = 𝛗₂ (M) are also two points on K(P).

M₁ is the tcP-Ceva conjugate of the gcP-isoconjugate of M and M₂ is the gcP-isoconjugate of the tcP-Ceva conjugate of M. With M = x : y : z, this gives :

M₁ = (a^2 (-c^2 x y-b^2 x z+a^2 y z)) / ((q+r) x) : : and M₂ = a^2 / ((q+r) x (q x+r x-p y-r y-p z-q z)) : : .

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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 cicum-cubic of the antimedial triangle passing through H (giving K028), X(20) (giving K443) and X(2889). The complement of this cubic is a circum-cubic 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).