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X(3), X(6), X(370), X(1249), five mates of X(370) (blue points), see Table 10 A', B', C' : midpoints of ABC traces of the trilinear polar of O 

Q099 is the locus of points whose polar line with respect to Q034 passes through the circumcenter O of ABC. Q099 is a quartic passing through X(370) and its five mates since the (light blue) tangents at these points to Q034 pass through O. The tangents at A, B, C are the altitudes. See the related quartic Q033. 
