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Let Q = p:q:r be a fixed point and M a variable point. The locus of M such that the trilinear polar of M is perpendicular to the line QM is a circumcubic TC(Q) passing through G, Q, the vertices Qa, Qb, Qc of the pedal triangle of Q and the points at infinity of the Thomson cubic. TC(Q) is also the locus of the common points of a line through Q with the circumconic passing through G and the trilinear pole of the perpendicular at Q to this line. This gives an easy construction of TC(Q). TC(Q) has equation : 

All these cubics form a net of K0 cubics i.e. the equation has no term in xyz. This net is generated by the three nodal cubics TC(A), TC(B) and TC(C). TC(A) has node A with two nodal tangents which are the bisectors at A. It contains the foot of the altitude AH and the two common points of the circumcircle and the perpendicular at G to BC. Obviously, TC(A) contains G and the points at infinity of the Thomson cubic. *** TC(Q) meets the circumcircle at A, B, C and three other points which also lie on pK(X6, X) where X is the point with first barycentric coordinate : a^2[3b^2c^2p+c^2q(a^2c^2)+b^2r(a^2b^2] X lies on the parallel at Q to the polar line of Q in the Stammler hyperbola. TC(Q) also contains Ra = AG /\ QQa and Rb, Rc defined similarly. 

Pivotal TC(Q) TC(Q) is a pK if and only if QaQbQc (triangle pedal of Q) is a cevian triangle i.e. if and only if Q lies on the Darboux cubic. In this case,
The following table gives a selection of such pivotal TC(Q). 



Nonpivotal TC(Q) TC(Q) is a nK (in fact a nK0) if and only if Qa, Qb, Qc are collinear i.e. either :
The following table gives a selection of such nonpivotal TC(Q). 



TC(Q) with concurring asymptotes TC(Q) is a K0+ if and only if Q lies on the line X(6)X(376). The asymptotes concur at a point on the line GK. All these cubics form a pencil which contains two nondegenerated K0++ : K314 = TC(X6) and K315 = TC(X376). These two cubics are central cubics with centers X(6), X(2) respectively.
Singular TC(Q) TC(Q) is singular if and only if Q lies at infinity (in which case it decomposes) or on a 10th degree curve passing through A, B, C, G, X(182), X(390). TC(G) = K295 has a double point at G, TC(X182) = K281 at K, TC(X390) at X(7).
TC(Q) through one or two given point(s) Let S be a given finite point distinct of G. TC(Q) passes through S if and only if Q lies on the perpendicular at S to the trilinear polar of S. Consequently, TC(Q) passes through two given finite points distinct of G if and only if Q is the intersection of two such perpendiculars. For example, for any Q on the Euler line TC(Q) contains H and for any Q on the Brocard line TC(Q) contains K hence TC(Q) contains H and K if and only if Q = O and then TC(Q) is the Thomson cubic. 
