Largest square inscribed in a rectangle The side length of the largest square inscribed in a rectangle of width $a$ and height $b$ is surely $\min(a,b)$. How is this proved? (Or if it be wrong, what is the correct result?)

The side length of a square is equal to the length of the diameter of the inscribed circle (which touches the square at the midpoints of its four sides). One diameter of the circle is parallel to the short side of the rectangle, hence cannot be longer than that side, or else the circle (and thus the square it's inscribed in) would lie partly outside the rectangle.
