Let ABC be a triangle, P a point, and (Op) the
cevian circle of P (with center Op).
Which is the locus of P such that the Poncelet point of Op
(ie the point of concurrence of the NPCs of OpBC, OpCA,
OpAB) lies on the cevian circle (Op) ?
Obviously G, H are on the locus.
For P = I, we have that the circles
Cevian_I, Pedal_I, NPC_ABC concur at Feuerbach point,
and the circles NPC_ABC, NPC_OpBC, NPC_OpCA, NPC_OpAB
concur at Poncelet point of Op
If the Poncelet point of Op lies on (Op) [= the cevian circle
of I], then the Poncelet point of Op coincides with
the Feuerbach point.