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).
Pick up a pencil, do some drawing, play with this. Look at angles APM, MPD, AEM, MCD and look for cyclic quadrilaterals. The proof that the lines AD and BE produced pass through P takes three or four lines.
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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?
Quadarc
Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.