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.

Proof
From Evolute of Cycloid is Cycloid, the evolute of a cycloid is another cycloid.

From Curve is Involute of Evolute, the involute of a cycloid is another cycloid as well.

But by the definition of involute, the path defined by the pendulum as described is the involute of the cycloid.

Hence the result.