What is this figure about?

Question

I saw this figure on first page of a Geometry book. I wonder what it wants to say. I understand these are medians of triangle ABC and I guess ACB',ABC',BCA' are equilateral triangles but I do not understand what is the point of this figure?

Answer 1

I think this is an illustration of Kiepert's solution of Lemoine's problem: find the vertices of a triangle $ABC$, given the outer vertices $A'B'C'$ of the equilateral triangles constructed on the sides of $ABC$.

One can show that $ABC$ are the midpoints of $A'A''$, $B'B''$, $C'C''$, where $A''B''C''$ are the outer vertices of the equilateral triangles constructed on the sides of triangle $A'B'C'$.