[math-fun] No Calculus, please!

I've long been a fan of the Buchardt (sp?) & Moser Scripta Math paper, No Calculus, Please! (I can't locate it on the internet and my copy is on loan to a grade 11 student) Non-calculus methods elude me for the moment for solving the following problem given me by the aforementioned student. What is the least length of hypotenuse of a right triangle which encloses an a by b rectangle? Astroid addicts will know that the answer is [a^{2/3} + b^{2/3}]^{3/2} Can anyone find that, using little more than AM geq GM ? You're allowed to take the dimensions of the rectangle to be c^3 by d^3 and get the min square on the hyp to be (c^2 + d^2)^3 --- if that helps. R.