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Crescents and Triangles
Age 14 to 16
Challenge Level 
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'.
Can you show that the sum of the areas of the two crescents equals the area of triangle ABC?
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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
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.