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

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?

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.

Circumspection

Age 14 to 16 Challenge Level

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.