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AD/ MD + BE/ ME + CF/ MF >= 9/2 , M interior, circumcircles related

Source: 2017 Croatia MO p7

August 3, 2020
geometrycircumcirclegeometric inequality

Problem Statement

The point MM is located inside the triangle ABCABC. The ray AMAM intersects the circumcircle of the triangle MBCMBC once more at point DD, the ray BMBM intersects the circumcircle of the triangle MCAMCA once more at point EE, and the ray CMCM intersects the circumcircle of the triangle MABMAB once more at point FF. Prove that holds ADMD+BEME+CFMF≥92\frac{AD}{MD}+\frac{BE}{ME} +\frac{CF}{MF}\ge \frac92
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