We start with the following problem:
Let $a$ and $b$ be positive integers such that their geometric and quadratic means are integers. Show that $a=b$.
One possible approach is to write down the corresponding diophantine equations and to do infinite descent on $u^4-v^4=w^2$ where $w=a^2-b^2$ which then has to be zero.
However, the means also have simple geometric representations. So my question is if there is a way to do the infinite descent geometrically.