IMAS 2019–2020 Junior Division Round 2 Question 15

In the quadrilateral $ABCD$, $AB\parallel CD$, point $E$ is the intersection of diagonal lines $AC$ and $BD$, and point $F$ is the midpoint of $AD$. Extend $EF$ to intersect $BC$ at point $G$ and let $A_1$, $B_1$, $C_1$ and $D_1$ be the feet of perpendiculars from $A$, $B$, $C$ and $D$ to line $FG$, respectively.
(1) Prove $\frac{AA_1}{CC_1}=\frac{BB_1}{DD_1}$; (5 marks)
(2) Prove $FE\ge GE$. (15 marks)