Let P, P* be two isogonal points; A'B'C' the pedal triangle of P*; Pa, Pb, Pc the isogonals of P in  AB'C', BC'A', CA'B'. The circles {PaB'C'}, {PbC'A'}, {PcA'B'} have a common point, and the same happens with the circles {A'PbPc}, {B'PcPa}, {C'PaPb}.