O is the circumcenter of acute triangle ABC, H is the Orthocenter. AD is perpendicular to BC, EF is the perpendicular bisector of AO,D,E on the BC. Prove that the circumcircle of triangle ADE passes through the midpoint of OH.

Let a, b, and c be real numbers greater 
than zero. define an xy coordinate system
such that A=(0,a), B=(-b,0), C=(c,0),
D=(0,0), E=(e,0), F=(f,g), H=(0,h),
K=(k,0), L=(i,j), M=(m,n), and P=(u,v).
 
Where F, K, L, M, and N are the midpoints
of line segments AO, BC, AB, OH, and AE
respectively.
 
It is easy, but tedious, to solve for all
the coordinates in terms of a, b, and c.
 
Then all that is needed is to verify that
 
   |PM|^2 = |PD|^2.
 

Posted by Bractals on 2013-02-03 16:16:12