Rotating Triangle

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

Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

A Chordingly

Stage: 3 Challenge Level:
Prove that the area of the annulus is ${1\over 4}\pi AB^2$ where $AB$ is a tangent to the inner circle.