# Fermat-Weber Point of a Polygonal Chain

Fermat-Weber Point of a Polygonal Chain

The Fermat–Weber point for a set of points in the plane is the point that minimizes the sum of the Euclidean distances from to the points of . A -chain of a regular -gon, denoted by (k), is the segment of the boundary of the regular -gon formed by a set of consecutive vertices of the regular -gon. Here we consider chains with an odd number of vertices whose middle vertex is . This Demonstration shows that for every fixed odd positive integer , as increases, the (blue) Fermat–Weber point of (k) moves on the axis toward the point on the boundary of the chain. It is also shown that when exceeds a certain integer, say , the (red) Fermat–Weber point of (k) coincides with the vertex of the chain lying on the axis.

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