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Rotating Triangle

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

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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?

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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Isosceles Interactivity

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
See the problem Isosceles.

The triangle OPA has a vertex O at the origin and OA along the x axis, such that P has coordinates (x, y) and A has coordinates (2x, 0). By moving the position of the point P infinitely many isosceles triangles can be formed all having an area of 12 square units. What is the locus of P such that the area of the triangle OPA is 12 square units? In the Java applet below, the point P is controlled by sliding the red point along the red line. If the red point disappears from view, simply press reload.