Answer to Question #247 in Geometry for Justin

Question

N points and the middles of all segments with endpoints in them are marked on the plane. Prove that the amount of marked points is not less than (2n - 3).

Answer 1

Let's consider that A and B are the most distant points from each other. Middles of segments, connecting point A (respectively, point B) with the remaining points, are all distinct and lie within a circle of radius AB/2 with the center in A (respectively, B). These two circles have only one common point, so the various amount of marked points is at least 2(n - 1) - 1 = 2n - 3.