Part 3/7:

Utilizing visual aids and geometrical concepts, one arrives at the conclusion that, instead of merely seeking an inscribed rectangle, it is more strategic to identify pairs of points on the loop where the line segments connecting them share the same midpoint and distance.

Mapping to a Higher Dimension

This reformulation leads to a three-dimensional representation of pairs of points from the loop, where each pair corresponds to a unique point in a three-dimensional space. The crux of the proof involves revealing that certain points on this focus surface must coincide, indicating the presence of an inscribed rectangle.