App:  Let D(0,0), A(0,a), B(b,0), C(c,0), M(0,m) be the coordinates of the points in question.  Since the triangle is acute a,c>0 and b>0 (WLOG).  Find the equations of the lines through DE and DF and see that they're opposites.

Entree:  Let a,b,c,d,e,f be complex numbers of absolute vale r that correspond to the vertices of the hexagon.  Show AOB=COD=EOF=pi/3 and so b = ae^(pi i/3) d = ce^(pi i /3), f = e e^(pi i /3) (the last one should make sense, I'm just too lazy to change the notation).  Let P,Q,R, be the midpoints of BC, DE, FA and show that RQ is obtained by rotating PQ around Q by 60 degress.

Dessert:  Show that the radii satisfy the recurrence R_1 = 1 R_{n+1} = R_n cos(pi/2^{n+1}).