Competition problem — Geometry — Maths Olympiad Prep

Geometry Difficulty 4.4 AIME Prove it Switzerland

Problem:

Let $ABC$ and $AMN$ be two similar, non-overlapping triangles with the same orientation, such that $AB = AC$ and $AM = AN$ . Let $O$ be the circumcentre of the triangle $MAB$ . Prove that the points $O$ , $C$ , $N$ and $A$ lie on a circle if and only if the triangle $ABC$ is equilateral.

Solution

Solution:

Let $O'$ be the circumcentre of triangle $NAC$ . Consider the rotation of center $A$ and angle $\angle BAC$ : the conditions of the problem imply that it maps $B$ to $C$ and $M$ to $N$ . Moreover, because it fixes $A$ we know that it maps $O$ to $O'$ , and triangle $AOO'$ is also similar to triangle $ABC$ . Then

$O, C, N, A$ on a circle $\Longleftrightarrow O'O = O'A \Longleftrightarrow CB = CA \Longleftrightarrow \triangle ABC$ equilateral, as desired.

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 reproduced verbatim; metadata (topic, difficulty) added by this project.