Since we have
\begin{equation}
\left( \frac{\lambda - \sqrt{a^2 + b^2}}{\sqrt{8}} \right)^2
= \frac{1}{2} \left( \frac{a + b}{2} \right)^2
\end{equation}
this result is a direct corollary to
R5209: Of all rectangles with a given perimeter, the square maximizes the area. $\square$
