It seemed to me that there might be a body of math work on this, since the geometric ratio for the sides, after all, is SQRT(phi) (the golden ratio).  I'll be darned if I could figure out how to construct the triangle, so I cheated and found this:

Seems that much better minds than mine solved this centuries ago!

Kudos to Ady for describing the problem in such a way that created only limited attention to the way this seems to be more commonly named