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The Fourth Intersection Point of the Circumconic

Construct a circumconic with center P. Let A', B', and C' be the antipodal points of A, B, and C. Circles AB'C', BA'C' and CA'B' intersect at the fourth intersection point T of the circumconic and the circumcircle. Circles ABC', ACB', and BCA' intersect at the antipodal point T'