You may also like

problem icon

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

problem icon


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?

problem icon


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.

The Medieval Octagon

Age 14 to 16 Challenge Level:


Medieval stonemasons used this method to construct exact octagons in a given square window.

Open your compasses to a radius of half the diagonal of the square and construct an arc with centre one vertex of the square - mark the 2 points where the arc crosses the sides.

Do that for all 4 vertices of the square giving 8 points which are the vertices of an octagon. Is the octagon an exact regular octagon ? Proof please.

If you have a Java enabled browser you can experiment with the interactive version