Given two points \(a\) and \(b\), on a closed curve, it is always possible to draw a circle using length \(ab\) as diameter. This circle will intersect the closed curve at, at least two points. It is possible to move \(a\) and \(b\) along the curve so that the center of the circle is inside the closed curve and the circle insects more points on the curve.
Since the center of the circle is inside the close curve there can be pairs of intersection points (\(a'b'\)) on opposite side of the diameter \(ab\), between the circle and the curve. Make it so.
Then it is possible to rotate/move/adjust the diameter \(ab\), with both points still on the closed curve, that \(a'b'\) is rotated to perpendicularly bisect the diameter \(ab\). (Changing one variable at a time, in this case the angle between the lines \(ab\) and \(a'b'\).)
When \(ab\) is perpendicular to \(a'b'\), \(aa'bb'\) forms a square.
Good night.
