What if one angle is obtuse?
Then we can argue as follows. WLOG we assume
I think I have found the solution based on this website:
http://planetmath.org/encyclopedia/BoundOnAreaOfRightTriangle.html
Lets go:
According to Heron's formula (http://en.wikipedia.org/wiki/Heron%27s_formula)
So, we are going to try to maximize the are of the triangle, subject to k = a^2 + b^2 + c^2 being constant, using Lagrange multipliers.
It is easier to work with S^2, and maximize it, so we set:
I felt lazy to solve the system of equations myself, so I plugged it into Mathematica (http://www.quickmath.com/webMathematica3/quickmath/page.jsp?s1=equations&s2=solve&s3=advanced).
The only valid solution with all a, b, c > 0 yields:
Therefore, and since we were trying to maximize S, we have:
Hope it helps!
Jose