Circle properties and circle theorems

problem

Medallions

Age

14 to 16

Challenge level

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

Circumspection

Age

14 to 16

Challenge level

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.
problem

Strange Rectangle

Age

16 to 18

Challenge level

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.
problem

Circumnavigation

Age

14 to 16

Challenge level

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

Flower

Age

16 to 18

Challenge level

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.
problem

Circle Scaling

Age

14 to 16

Challenge level

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

Sports Equipment

Age

7 to 11

Challenge level

If these balls are put on a line with each ball touching the one in front and the one behind, which arrangement makes the shortest line of balls?
problem

Coins on a Plate

Age

11 to 14

Challenge level

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.
problem

Dodecagon Angles

Age

11 to 14

Challenge level

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

ArRh!

Age

14 to 16

Challenge level

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?