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A"B"C" is equilateral

Source: India tst 2003, p1

March 14, 2012
geometrycircumcirclegeometry proposed

Problem Statement

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