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I need help understanding the proof of lemma 2.1. in these notes here. The proof and lemma are the following:

Lemma 2.1. Every point on a regular polygon is determined, among all points on the polygon, by its distances from two adjacent vertices of the polygon.

Proof. In the picture above, let the black dots be adjacent vertices of a regular polygon. Since they are adjacent, the line segment connecting them is an edge of the polygon and the polygon is entirely on one side of the line through the black dots. That shows the two blue dots can’t both be on the polygon, so a point on the polygon is distinguished from all other points on the polygon (not from all other points in the plane!) by its distances from the two adjacent vertices.

I understand the proof up to "..., so a point on the polygon is distinguished from all other points on the polygon..."

Please could someone help me understand how the fact that not both points can be on the polygon implies that every point on the polygon is uniquely determined by its distance from given two points?

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    $\begingroup$ Normal Human: I edited the title accordingly. $\endgroup$
    – a student
    Nov 3 '15 at 9:19
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It means that, if one of the two points stays in the polygon the other stays not on the polygon, so, given the two radii $r_1$ and $r_2$ of the circles, there can be only one point in the polygon that has such distances from the two vertices, and this is true for all points in the polygon.

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  • $\begingroup$ Apparently, it suffices for the polygon to be convex rather than regular $\endgroup$ Nov 3 '15 at 9:40

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