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collinearity and concyclic wanted, 2 intersecting circles

Source: 1991 Greece MO Grade XI p2

September 6, 2024
geometrycollinearConcyclic

Problem Statement

Given two circles (C1)(C_1) and (C2)(C_2) with centers O1\displaystyle{O_1} and O2O_2 respectively, intersecting at points AA and BB. Let ACAC και ADAD be the diameters of (C1)(C_1) and (C2)(C_2) respectively . Tangent line of circle (C1)(C_1) at point AA intersects (C2)(C_2) at point MM and tangent line of circle (C2)(C_2) at point A intersects (C1)(C_1) at point NN. Let PP be a point on line ABAB such that AB=BPAB=BP. Prove that: a) Points B,C,DB,C,D are collinear. b) Quadrilateral AMPNAMPN is cyclic.
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