In a triangle ABC,
H is the orthocenter and D is a point on the circumcircle O. D_{1},
D_{2}, and D_{3} are the reflections of D over BC, AC,
and AB, respectively. DH meets the Simson line for D at F. Prove that
(1) D_{2},
D_{1}, H, and D_{3} are collinear points; (2) D_{2}D_{3}
= 2.SM; (3) F is the midpoint of DH.
