If we embed the statement in the complex plane, the proof is a mere calculation. In the notation of the figure, we can assume that $A=0$, $B=1$, and $C$ is in the upper half plane. The hypotheses are

where $\alpha=\exp{\pi i/3}$, and the conclusion we want is

where

From (1) and the relation $\alpha^{2}=\alpha-1$, we get $X,Y,Z$:

and so

proving (2).

Remarks: The attribution to Napoléon Bonaparte (1769-1821) is traditional, but dubious. For more on the story, see MathPages.