E15. Area in a square

Given a square ABCD of side length a, let M be the midpoint of AB and N be the midpoint of BC. Draw AN and CM, and let their intersection be O. Find the area of AOCD.

I saw this problem on an Internet math forum, along with an solution involving Cartesian geometry. The solution was straightforward and not particularly difficult. However, I'm including the problem with a challenge to do it without introducing coordinates. I think the solution that I found is much prettier than the coordinate-based solution. What do you think?

Note: There is something wrong with scribd which is not allowing me to post my solution, which is a short pdf file. I will add a link later when I can upload my file.