centre and other points inside

Let ABC be a triangle with O the circumcenter and I the incenter. Let D,E and F be the tangent points of the inscripted circle with AB, BC and AC.
1. Prove that orthocenter of DEF lies on the line IO
2. Prove that the distance of D, E and F from IO are three numbers where the biggest among the three is equal to the sum of the other two.

I thinks that some property about points circle could help, but can't figure out how!...(Crying)

Hey pincopallino.

I've never heard of the incenter: can you explain what this is?

Incenter is the center of a circle inscribed in a triangle