FKMO 2020 Problem 1

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$ and $\overline{AD}=\overline{BC}, \overline{AB}>\overline{CD}$ . Let $E$ be a point such that $\overline{EC}=\overline{AC}$ and $EC \perp BC$ , and $\angle ACE<90^{\circ}$ . Let $\Gamma$ be a circle with center $D$ and radius $DA$ , and $\Omega$ be the circumcircle of triangle $AEB$ . Suppose that $\Gamma$ meets $\Omega$ again at $F(\neq A)$ , and let $G$ be a point on $\Gamma$ such that $\overline{BF}=\overline{BG}$ . Prove that the lines $EG, BD$ meet on $\Omega$ .