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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

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


Stage: 4 Challenge Level: Challenge Level:1

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?

Diagram of the problem