1866. A Property Of Kiepert Circum Hyperbola

 This problem is inspired from #5175 by Tran Quang Hung.

Let P,Q be two points on Kiepert hyperbola of ABC. Circles APQ, BPQ, CPQ intersects the Kiepert hyperbola of ABC at A', B',C' resp.
R=X(2)-Centroid of of A'B'C' lies on Kiepert hyperbola of ABC.
So we have a transformation T, T(P,Q)->R, Let's show it with * operation. Then we have:
P*Q=R, P*R=Q, Q*R=P

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