Two smaller circles, centres B and C touch each other externally and touch the circle centre A internally. What happens to the perimeter of triangle ABC as the two smaller circles roll around touching the rim of the bigger circle or as the two smaller circles vary in size?

If you have a Java-enabled browser you can view an interactive version of this problem below. You can move the red points around to adjust the diagram. Moving the point X on the vetrical slider controls the size of the smaller circle.