Generalization of Napoleon’s Theorem: Similar Triangles

Consider an arbitrary triangle ABC. Let us erect on its sides equilateral triangles $ABZ, BCX$ and $CAY$, further let these equilateral triangles be oriented in such a way that either none of them overlap with triangle $ABC$ (they are oriented outwards) or all of them overlap (they are oriented inwards). 

Then the centroids $L,M$ and $N$ of these triangles form an equilateral triangle.

Click on the image.