Here's a geometric proof.
If we try to construct a square equal in area to a given rectangle and the
dimensions of the rectangle are a and b, then the problem is to determine x
such that .
Look at the semi-circle in the diagram. The inscribed triangle is a right triangle, then the two smaller ones are similar. Then, we have:
Therefore, x is mean proportional between a and b, so there geometric mean is .
Since the radius of the semi-circle is , it follows that:
This holds even when a=b.
The geometric mean of two positive numbers never exceeds their arithmetic mean
Another thing to look at algebraically is since
Thanks for the proofs - they were very helpful.
Does anyone know of a good book that explains/gives practice with proofs? I don't need to for any of my courses (I'm currently in Calc II), but I think proofs are interesting, and I'd like to get better at them. Sadly I have little experience because none of the math courses I've taken required me to do proofs. Even when I took geometry in high school, they dumbed it down and barely required any proofs. It's kind of unfortunate, because I'd have more mathematical discipline if it was for a grade. Anyway, I would be interested to learn about proofs in my spare time.