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two triangles have equal circumradii

Source: Italy TST 2009 p5

March 10, 2012
geometrycircumcircletrapezoidgeometric transformationreflectionpower of a pointradical axis

Problem Statement

Two circles O1O_1 and O2O_2 intersect at M,NM,N. The common tangent line nearer to MM of the two circles touches O1,O2O_1,O_2 at A,BA,B respectively. Let C,DC,D be the symmetric points of A,BA,B with respect to MM respectively. The circumcircle of triangle DCMDCM intersects circles O1O_1 and O2O_2 at points E,FE,F respectively which are distinct from MM. Prove that the circumradii of the triangles MEFMEF and NEFNEF are equal.
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