Points symmetric with respect to perpendicular bisector
Source: Problem 6 of Russian Regional Olympiad 2011, grade 9
September 1, 2011
geometrycircumcircleperpendicular bisectorgeometry proposed
Problem Statement
Initially, there are three different points on the plane. Every minute, three points are chosen, for example , and , and a new point is generated which is symmetric to with respect to the perpendicular bisector of line segment . 24 hours later, it turns out that among all the points that were generated, there exist three collinear points. Prove that the three initial points were also collinear. (Author: V. Shmarov)