Question 63
Let $D$ be a point on the side $[BC]$ of the triangle $ABC$ different from the vertices and $E$ be the midpoint of $[CD]$. The line perpendicular to $BC$ at $E$ intersects the side $[AC]$ at point $F$ satisfying $AF \cdot BC = AC \cdot EC$. Let $G$ be the second point where the circumcircle of the triangle $ADC$ intersects the side $[AB]$. Prove that the tangent line of the circumcircle of the triangle $AGF$ at $F$ is also tangent to the circumcircle of the triangle $BGE$.