This problem is very special , there is a beautiful theorem behind this , see if you can discover it !

Problem :

Outwards along the sides of convex quadrilateral $\displaystyle ABCD$ are constructed equilateral triangles $\displaystyle WAB , XBC , YCD , ZDA$ with centroids $\displaystyle S_1 , S_2 , S_3 , S_4$ , respectively . Prove that $\displaystyle S_1 S_3$ is perpendicular to $\displaystyle S_2 S_4$ if and only if $\displaystyle AC = BD$ .

( Mediterranean Mathematical Competition 2000 )

In fact , equilateral triangles can be changed to squares with centers instead of centroids .