Two congruent equilateral triangles and in the plane are given. Show that the midpoints of the segments either are collinear or form an equilateral triangle.
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Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.
Izvor: Međunarodna matematička olimpijada, shortlist 1971
Školjka 
and 
in the plane are given. Show that the midpoints of the segments 
either are collinear or form an equilateral triangle.