Normal to Cycloid passes through Bottom of Generating Circle

Theorem
Let $C$ be a cycloid generated by the equations:
 * $x = a \paren {\theta - \sin \theta}$
 * $y = a \paren {1 - \cos \theta}$

Then the normal to $C$ at a point $P$ on $C$ passes through the bottom of the generating circle of $C$.