Pendulum Contained by Cycloid moves along Cycloidal Path

Theorem
Let a pendulum with a flexible rod be suspended from a point $P$.

Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs.


 * CycloidPendulum.png

Then the bob is constrained to move such that its path traces the arc of a cycloid.