In the figure below, given a triangle
ABC, circumcircle C_{0}, circumradius R, line DEF
parallel to AC and line FGM parallel to AB. C_{1}, C_{2},
and C_{3}, and R_{1}, R_{2}, and R_{3}
are the circumcircles and circumradii of triangles DBE, FGE, and
MGC respectively, prove that: R // R_{1} // R_{2}
// R_{3}, and circles C_{0} and C_{1}
are tangent at B, circles C_{1} and C_{2} are
tangent at E, circles C_{2} and C_{3} are
tangent at G, and circles C_{3} and C_{0} are
tangent at C.
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