Baltic Way 2011 · Shortlist problem

Geometry

Let $\Gamma$ be a circle, and $A$ a point outside $\Gamma$ . For a point $B$ on $\Gamma$ , let $C$ be the third vertex of the equilateral triangle $ABC$ (with vertices $A$ , $B$ and $C$ going clockwise). Find the path traced out by $C$ as $B$ moves around $\Gamma$ .

Change pool

When you’re ready

Review material becomes available with the next Daily.

Review

Topics

Angles and distances · Constructions, loci, concurrency and collinearity · Transformations

Solutions

Solution

Let $\Gamma$ have centre $O$ and radius $r$ . Let $O'$ be the third vertex of the equilateral triangle $AOO'$ (clockwise). Then the locus of $C$ is a circle with centre $O'$ and radius $r$ : for consider the triangles $AOB$ and $AO'C$ . Then $AO = AO'$ and $BA = CA$ (by construction: equilateral triangles) and $\angle BAO = 60^\circ - \angle O'AB = \angle CAO'$ . So the two triangles are congruent (two sides and an included angle) and hence $BO = CO'$ . Thus $C$ lies on a circle with centre $O'$ and radius $r$ .