I was and still am so glad to get out of school but constructions seem to drag me back there- which is why I try to avoid them. But this one is really interesting and is fun to do:
You are given a sheet of paper with two intersecting circles G1 and G2 with unknown centres. They meet at points C and D. Find their centres using only a straightedge.
A straightedge is a ruler without markings that allows you to join two points by drawing a line through them.
It might help to solve these problems here:
Here is how it was done by me:
Here E, F, M and N are the points that I chose. MN and it's parallel line are marked in green and EF and it's parallel line are marked in blue. K and S are the points where the nonparallel sides of the isosceles trapeziums meet. And the lines marked in red are the ones that intersect at the centre U of one of the circles.
I've explained the method in a reply to Michael Mendrin in the comments below.