2D shapes and their properties

Explore the geometry of these dart and kite shapes!

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.

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

We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point D?

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?

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

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

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

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