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.

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