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FN > ND, excircle of ABC and circumcircle of CDM related

Source: Ukraine TST 2010 p11

May 5, 2020
geometrycircumcirclegeometric inequalityexcircle

Problem Statement

Let ABCABC be the triangle in which AB>ACAB> AC. Circle ωa\omega_a touches the segment of the BCBC at point DD, the extension of the segment ABAB towards point BB at the point FF, and the extension of the segment ACAC towards point CC at the point EE. The ray ADAD intersects circle ωa\omega_a for second time at point MM. Denote the circle circumscribed around the triangle CDMCDM by ω\omega. Circle ω\omega intersects the segment DFDF at N. Prove that FN>NDFN > ND.
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