Is there a way to prove the property of the parabola that a ray parallel to its axis of symmetry if "reflected" by the parabola will intersect the symmetry axis of the parabola, without using analytic geometry?
Assuming that $DA\parallel CB$ and that the line has been reflected,I want to prove that B is in fact the focus of the parabola.
Let $F$ be focus of parabola.
It is well known that if we project orthogonal any point $P$ on parabola to it directrix we get such point $P'$ on directrix, that perpendicular bisector $t$ of $FP'$ is tangent of parabola at $P$. Thus if we reflect $PG$ across $t$ we get parallel through $P$ to axis of symmetry of parabola.
So if we reverse this process, we see that if reflect parallel to axis of symmetry at point $P$ on parabola, this reflection must go through $F$.