Divide a circular disk into seven parts with a straightedge and compass such that each part has the same area and perimeter.

(In reply to Heptagon vs. 180 rotation by nikki)

The idea is that the probability is zero that the seventh mark will ever exactly land on the initial tick mark. Of course in the real world, nothing will be exact anyway, but the particular rules of geometric constructions assume that you can match a distance with a compass setting exactly and make an exact circle with the compass.

On the diagram pointed to by Jer's post, imagine first drawing that line out from the left end of the diameter and going up and to the left.  Mark off 6 arbitrarily long (but equal) distances.  These points each have a particular distance from the left point and the right point.  Match the compass to each of these distances in turn, and make circular arcs with those lengths down and to the right from the right end of the diameter.  By SSS, where the arcs meet, the formed triangles will be congruent, and the points will be located at the same angle down and to the right as the originals were up and to the left.

Posted by Charlie on 2004-12-10 21:57:34