A Parallel Line Turns Tangent

The parabola $C:y^2=2px$ ($p>0$) has focus $F$, and its directrix meets the $x$-axis at a point $D$. A line through $F$ meets $C$ at two points $A$ and $B$, with

1. Find the equation of the parabola $C$. 2. Let $P$ and $Q$ be distinct points on $C$ with $PF\perp x$-axis. The line $PQ$ meets the $x$-axis at a point $G$, and a point $E$ is chosen on the $x$-axis with $|GE|=|GD|$ and $G$ between $E$ and $D$. Let $l$ be the line through $Q$ parallel to $PE$. Prove that $l$ is tangent to the parabola $C$.