A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle

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

Arrh!

Stage: 4 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.