Circle properties and circle theorems

Weekly Problem 50 - 2012
The diagram shows a regular dodecagon. What is the size of the marked angle?

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

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.

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.

An equilateral triangle is sitting on top of a square. What is the radius of the circle that circumscribes this shape?

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.

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

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

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

Describe how to construct three circles which have areas in the ratio 1:2:3.