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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

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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?

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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:
rectangles and crescents

Semicircles are drawn on the sides of a rectangle ABCD as shown in the diagram. A circle passing through points ABCD carves out four crescent-shaped regions (coloured yellow and green in the diagram).

Prove that the sum of the areas of the four crescents is equal in area to the rectangle ABCD.