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X(3) 

An inversible cubic iK is symmetric in one of the perpendicular bisectors if and only if its pivot is the reflection of one vertex of ABC in the corresponding bisector. Hence, there are three such cubics presenting a high number of invariances : isoconjugation, 3 inversions, one axial symmetry. The figure above shows K111A symmetric in the perpendicular bisector of BC. 
