MathNet 2018-part-01 — 181 problems (EN–VI) Question 106

The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of the triangle $CMD$ intersects the line $AC$ at $C$ and $E$. The circumcircle of the triangle $AME$ intersects the line $AB$ at $A$ and $F$.
Prove that $CF$ is an altitude of the triangle $ABC$.