IN-TST 2003 Problem 1

Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$ , respectively, of an acute non-isosceles triangle $ABC$ , and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$ . Define analogously the points $B''$ (on arc $EB'$ ) and $C''$ (on arc $FC'$ ). Show that triangle $A''B''C''$ is equilateral.