## Proof of Theorem 3.1

1. The polygon is convex.
This follows from the way the polygon is constructed.

2. The distance between p5 and p1, p2, p3 and p8 is greater than the distance between p6 and p1, p2, p3 and p8, respectively.
Since the polygon is convex, then by Lemma A.3 line segments p1p5, p2p5, p3p5 and p8p5 intersect line segment p6p3. The polygon construction states that p6 lies inside the circle centered at p3 with radius d = d (p3, p5). Therefore, if we let x = p6, u = p3 and v = p5 and let p1, p2, p3 and p8 take their turns being w then by Lemma A.2 statement 2 is true.

3. The distance between p1 and p4, p5, p6, p7 and p8 is greater than the distance between p2 and p4, p5, p6, p7 and p8, respectively.
Since the polygon is convex, then by Lemma A.3 line segments p4p1, p5p1, p6p1, p7p1 and p8p1 intersect line segment p2p8. The polygon construction states that p2 lies inside the circle centered at p8 with radius d (p8, p1). Therefore, if we let x = p2, u = p8 and v = p1 and let p4, p5, p6, p7 and p8 take their turns being w then by Lemma A.2 statement 3 is true.

4. The diameter of the polygon is achieved only by p3 and p7.
We leave the proof of this statement to the reader. Most cases follow directly from statements 2 and 3. The key is to remember that d (p3, p7) > d where d is the side length of triangle p1p3p5.

Back to Theorem 3.1...

Matthew Suderman
Cmpt 308-507