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?

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.

Three Four Five

Stage: 4 Challenge Level:

As shown in the diagram, two semi-circles (each of radius $1/2$)
touch each other, and a semi-circle of radius 1 touches both of
them. Find the radius of the circle which touches all three
semi-circles, and find a 3-4-5 triangle in the diagram.