Take a strip of paper and make a knot.
A pentagonal shape is formed by the edges of the paper strip.
If the ends of the strip are folded along the edges of the strip the result looks like a regular pentagon.
Show that, the resulting figure IS a regular pentagon.
HINT: Use a paper strip to fold the knot. Now carefully unfold it and examine the pattern of parallel lines and transversals that the crease denote. One path to a proof can make use of these patterns.
Hint: Note that the pattern when the knot is unfolded shows congruent trapezoids, each isosceles with the two slant sides and the shorter base being the same length. The ratio of the two parallel bases is the Golden Ratio.