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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 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? Circumspection

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.

The Pillar of Chios

Age 14 to 16 Challenge Level: The whole shape is made up of a rectangle, two semicircles on $AB$ and on $DC$ together making one circle, and the two semicircles on $AD$ and $BC$ making another circle.

Excellent solutions were sent in by a pupil from Dr Challoner's Grammar School, Amersham and Nisha Doshi ,Y9, The Mount School, York. Here is one of their solutions: Take: $AB=2x, AD=2y$.
\begin{eqnarray} \mbox{Total area of shape} &=& \pi x^2 + \pi y^2 + (2x \times\ 2y)\\ &=& \pi x^2 + \pi y^2 + 4xy. \end{eqnarray}
By Pythagoras Theorem
\begin{eqnarray} AC^2 &=& AD^2 + DC^2\\ &=& (2x)^2 +\ (2y)^2\\ &=& 4(x^2 + y^2)\\ \mbox{Area of big circle} &=& \pi(\mbox{AC}/2)^2\\ &=& \pi(x^2 + y^2)\\ \mbox{Area of crescents} &=& \mbox{Area of shape - Area of big circle}\\ &=& \pi x^2 + \pi y^2 + 4xy - \pi (x^2 + y^2)\\ &=& 4xy\\ \mbox{Area of rectangle} &=& 2x \times\ 2y\\ &=& 4xy \end{eqnarray}
so the sum of the areas of the four crescents is equal in area to the rectangle $ABCD$.