Circle properties and circle theorems

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

Keep constructing triangles in the incircle of the previous triangle. What happens?

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?

Can you find a relationship between the area of the crescents and the area of the triangle?

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?