Competition problem — Geometry — Maths Olympiad Prep

Library / /1 of 26

Geometry Difficulty 5.4 AIME, harder Prove it Russia

Five points are marked in the plane. It is permitted to choose some of them and change their positions; the distances between the chosen points should be preserved. Prove that it is possible to perform such change so as to obtain a configuration of five points possessing an axis of symmetry.

Solution

Let us denote the given points by $A$ , $B$ , $C$ , $D$ , and $E$ . Choose among them two points that are the farthest apart; let these be $A$ and $B$ . We will show that it is possible to move them as required.

Draw the perpendicular bisector $\alpha$ to the segment $CD$ . If $E$ lies on $\alpha$ , then it is sufficient to move points $A$ and $B$ onto the line $\alpha$ . Otherwise, let $E'$ be the point symmetric to $E$ with respect to $\alpha$ . Note that the distance from $E$ to $\alpha$ is less than the length of one of the segments $EC$ or $ED$ , that is, less than $AB$ .

Thus, we can move point $A$ to point $E'$ , and point $B$ to a point lying on $\alpha$ and at a distance $AB$ from $E'$ . It is easy to see that $\alpha$ is the axis of symmetry of the new set of points, as required.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.