Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

Running Race

Stage: 4 Short Challenge Level:

All three runners finish at the same time.

Let the radius of $R$'s track be $r$ and let the radius of the first semicircle of $P$'s track be $p$; then the radius of the second circle of this track is $r-p$.

The total length of $P$'s track is $\pi p + \pi(r-p) = \pi r$, the same length as $R$'s track.

By a similar argument, the length of $Q$'s track is also $\pi r$.

This problem is taken from the UKMT Mathematical Challenges.