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Problem 55) Let ABC be a triangle. Incircle (I) touches BC, CA, AB at D, E, F. M is a point on circle center A which passes though E, F.
a) Prove that pedal triangle XYZ of M wwith respect to triangle DEF is right triangle.
b) DM cuts IA at K. MI cuts EF at T. Prove that K lies on circumcircle (DEF) if only if T lies on circumcircle (XYZ).
c) M' is isogonal conjugate of M with respect to triangle DEF. Prove that M' always lies on fixed circle.
Problem 56) Let ABC be a triangle and point P. A'B'C' is pedal triangle of P with respect to triangle ABC. O is circumcircle of triangles ABC, (O') is circumcircle of triangle A'B'C'. PA', PB', PC' intersects (O') again at A1, B1, C1, repectively. Assume that P, O O' are collinear. Prove that circumcircles (PAA1), (PBB1), (PCC1) have a common point other than P.
Problem 57) Let ABC be a triangle with circumcircle (O). A circle (K) pass though B, C intersects AB, AC at F, E, respectively. O1, O2 are circumcenter of triangle ABE, ACF, respectively. (L) is circumcircle of triangle KO1O2. P is point on (L). The line passes though P and perpendiculer to OP intersects (O) at B', C'. Prove that nine-point center of triangle AB'C' always lies on a fixed circle (J) and LJ perpendiculer to EF.