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?

Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

Curvy Areas

Stage: 4 Challenge Level:
Suppose the radius of the smallest semicircle is $x$.
What are the radii of the other semicircles?
What areas can you work out, in terms of $x$ and $\pi$?