Cycloid has Tautochrone Property

Theorem
Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line.

Let a bead $B$ be released from some point on the wire.

The time taken for $B$ to reach the lowest point of $C$ is:

independently of the point at which $B$ is released from.

Also see

 * Brachistochrone is Cycloid, in which it is shown that a cycloid is also the shape for which the time is shortest.