Solution:
- Let L be the line of symmetry for the simple closed curve J. We may take the direction of L to be horizontal. In fact, we can assume that L is the x-axis if we wish.
- Then L and J will have two points, say P and Q in common. Furthermore, there is some point R on the curve such that the distance from R to L (the y-coordinate of R) is maximum.
- Use the "mountain climbing" technique from Theorem B to have two climbers begin respectively at P and Q and each move along the "upper" half of J so that at each instant their y-coordinates are equal. (As we noted in the proof outline of Theorem B, there's some messy work to show that this can be done, but it does work.)
- Note that at any time, the two climbers along with their "reflections" in the "lower" part of the curve J form a rectangle.
- The rectangle formed starts very wide and short, and ends (as the climbers approach R) skinny and tall. At some time along the way, it must have been a square!
Note: We have actually proved that any simple closed curve symmetric across a line has an inscribed square with two sides parallel to the line of symmetry.