Coins on a Plate

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.

Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?

Semicircle in a Semicircle

Weekly Problem 2 - 2008
The diagram shows two semicircular arcs... What is the diameter of the shaded region?

Not So Little X

Stage: 3 Challenge Level:

Congratulations to Matthew Hodgetts, King Edward VI Camp Hill School, Birmingham, to Suzanne Abbott, Nisha Doshi and Christiane Eaves, Mount School, York; to Jessica Zhang; to the KS3 Maths Club, Strabane Grammar School and finally to Michael Swarbrick-Jones, Y7 Comberton Village College, Cambridge whose solution appears below.

x = the diameter of one of the circles.
Since (AC) and (BD) are both the length of a radius then x = (AC) + (BD).

So x = 12 - (AB) = 12 - x/3.
So x + x/3 = 12
or 4/3 x = 12.

To find x we take the reciprocal of 4/3 and multiply it by 12.
x = 9 units.