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

Ok, I have a question.  How exactly do you make the exactly 180 degree rotation of that line extension (that helps you divide the diameter into even sevenths)?  Remember you have a compass not a protractor.  The thoughts I have all have some reasonable error involved.

In that case, why can't I try making a heptagon by using trial and error to divide the circumference of the disk?  My process would be...

Make a tick mark on the circumference of the disk.  Guesstimate how far apart you should set you compass.  Starting at the tick mark, go around the disk marking off this compass length.  If your seventh mark doesn't overlap with the initial tick mark then adjust your compass accordingly.  (Note, I would erase all but the first marking.  That way if during my first trial my compass distance was too large, and during my second trial it was too small, then I will still have the mark from my first trial to show me my limits... I hope that made sense).

Once I finally get a trial where the seventh mark lands on the initial tick mark, I have the vertices of my heptagon.  Then I can easily find the center of the disk (several ways to do this... two chords, then two perp bisectors will find the center but that takes 4 lines and 4 curves.  Or there's this other way with 3 curves and 2 lines).

Connect each vertex to the center and we have 7 equal pie slices.

Just to be clear, I am not saying that Eric's solution isn't correct.  Indeed it is, and quite ingenious in my opinion.  I never heard of that method for dividing a line into x pieces.  It was just the "then draw a line from the other end point at a 180 degree rotation" that bothered me.  If someone would explain how you do that precisely then I will give up my heptagon idea.

Posted by nikki on 2004-12-10 21:07:21