TITLE: A specific example regarding the inscribed square problem
QUESTION [5 upvotes]: Toeplitz' conjecture (also called inscribed square problem) says that:

For every Jordan curve $\mathscr C$, there exists four distincts points $A$, $B$, $C$ and $D$ belonging to $\mathscr C$ such that $ABCD$ is a square.

A Jordan curve is a non self-intersecting continuous loop.
Here is a drawing to illustrate the situation, and a link to the Wikipedia page if you want to find out more about this conjecture.

The conjecture has already been proven in several cases, including when $\mathscr C$ is piecewise analytic.
So we know that for these two figures, there exists an inscribed square.



The question is how do I find those squares?

REPLY [5 votes]: There is. If we draw an orthogonal line with respect to the one that is aligned with the curve, then you can build the square.