Problem
GEO-B3-M05-P004 Centers of Three Equilateral Triangles
#4
★☆☆☆☆ Level 1 of 5
External equilateral triangles are constructed on the sides of \(ABC\). Prove that their centers form an equilateral triangle.
Inspired by Prasolov special points geometry method
C. Hint 1. Use \(60^\circ\) rotations.
D. Hint 2. Compare the segments between neighboring centers as images under rotation.
Let \(X,Y,Z\) be the centers of the equilateral triangles on \(BC,CA,AB\). Consider the \(60^\circ\) rotation associated with the equilateral triangle on side \(AB\). It sends the direction from one constructed center to another to the next such direction.
The cyclic argument gives \(XY=YZ=ZX\), and the angle between \(XY\) and \(YZ\) is \(60^\circ\). Hence \(XYZ\) is equilateral.
This is the starting form of Napoleon's theorem; more general versions can be proved later.