Answer to Question #233 in Geometry for Laura

Question

Polygon has the center of symmetry O. Prove that the sum of the distances to its vertices is minimal for point O.

Answer 1

If A and A’ are symmetrical about the point O vertices of the polygon then the sum of the distances to points A and A’ is the same for all points of the segment AA’. For all other points it is bigger. Point O belongs to all such segments.