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.

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?

Some(?) of the Parts

Age 14 to 16 Challenge Level:

A circle touches the lines $OA$ extended, $OB$ extended and $AB$ where $OA$ and $OB$ are perpendicular.

Show that the diameter of the circle is equal to the perimeter of the triangle.