VN-TST 1996 Problem 4

Given 3 non-collinear points $A,B,C$ . For each point $M$ in the plane ( $ABC$ ) let $M_1$ be the point symmetric to $M$ with respect to $AB$ , $M_2$ be the point symmetric to $M_1$ with respect to $BC$ and $M'$ be the point symmetric to $M_2$ with respect to $AC$ . Find all points $M$ such that $MM'$ obtains its minimum. Let this minimum value be $d$ . Prove that $d$ does not depend on the order of the axes of symmetry we chose (we have 3 available axes, that is $BC$ , $CA$ , $AB$ . In the first part the order of axes we chose $AB$ , $BC$ , $CA$ , and the second part of the problem states that the value $d$ doesn't depend on this order).