You may also like

problem icon

Some(?) of the Parts

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

problem icon

Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

problem icon

Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Rotating Triangle

Stage: 3 and 4 Challenge Level: Challenge Level:1

Edward from Graveney School, Tooting, London sent this solution.
Let us make $a$ the radius of the largest circle centre $A$ etc. Then the lengths of the sides of the triangle are: $AB = a - b$, $AC = a - c$ and $BC = b + c$.
The perimeter of the triangle is: $$AB + BC + CA = (a - b) + (a - c) + (b + c)= 2a.$$ So the perimeter of the triangle is twice the radius of the large circle whatever the sizes of the small circles.