Regular polygons and circles

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.

Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?

What fractions of the largest circle are the two shaded regions?

This pattern of six circles contains three unit circles. Work out the radii of the other three circles and the relationship between them.

Make an estimate of how many light fittings you can see. Was your estimate a good one? How can you decide?

Make different quadrilaterals on a nine-point pegboard, and work out their angles. What do you notice?

Weekly Problem 1 - 2006
The diagram shows two circles enclosed in a rectangle. What is the distance between the centres of the circles?

Investigate these hexagons drawn from different sized equilateral triangles.