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Proof of Theorem 3.1

*1. The polygon is convex.*

This follows from the way the polygon is constructed.

*2. The distance between **p*_{5} and *p*_{1}, *p*_{2}, *p*_{3} and *p*_{8}
is greater than the distance between
*p*_{6} and *p*_{1}, *p*_{2}, *p*_{3} and *p*_{8}, respectively.

Since the polygon is convex,
then by Lemma A.3
line segments *p*_{1}*p*_{5}, *p*_{2}*p*_{5}, *p*_{3}*p*_{5} and *p*_{8}*p*_{5} intersect
line segment *p*_{6}*p*_{3}.
The polygon construction states that *p*_{6} lies
inside the circle centered at *p*_{3} with radius
*d* = *d* (*p*_{3}, *p*_{5}).
Therefore, if we let *x* = *p*_{6}, *u* = *p*_{3} and *v* = *p*_{5}
and let *p*_{1}, *p*_{2}, *p*_{3} and *p*_{8}
take their turns being *w* then
by Lemma A.2 statement 2 is true.

*3. The distance between **p*_{1} and *p*_{4}, *p*_{5}, *p*_{6}, *p*_{7} and *p*_{8}
is greater than the distance between
*p*_{2} and *p*_{4}, *p*_{5}, *p*_{6}, *p*_{7} and *p*_{8}, respectively.

Since the polygon is convex,
then by Lemma A.3
line segments *p*_{4}*p*_{1}, *p*_{5}*p*_{1}, *p*_{6}*p*_{1}, *p*_{7}*p*_{1} and *p*_{8}*p*_{1} intersect
line segment *p*_{2}*p*_{8}.
The polygon construction states that *p*_{2} lies
inside the circle centered at *p*_{8} with radius
*d* (*p*_{8}, *p*_{1}).
Therefore, if we let *x* = *p*_{2}, *u* = *p*_{8} and *v* = *p*_{1}
and let *p*_{4}, *p*_{5}, *p*_{6}, *p*_{7} and *p*_{8}
take their turns being *w* then
by Lemma A.2 statement 3 is true.

*4. The diameter of the polygon is achieved only by **p*_{3} and *p*_{7}.

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* (*p*_{3}, *p*_{7}) > *d*
where *d* is the side length of triangle *p*_{1}*p*_{3}*p*_{5}.

Matthew Suderman

Cmpt 308-507