Quadrilaterals + Ptolemy's Theorem + Triangle proof

Hi I really need help in a few problems, any help would be greatly appreciated. I attempted each problem but had no suscess in deriving anything useful =\.

1) Prove that a trianglke is equilateral if and only if its medians are congruent.

2*) In triangle ABC, prove that the vertex A, the orthocenter H, and the feet of altidudes from B and C are concyclic. Orhtocenter H is inside Triangle ABC.

3***) Find the distance between the vertices at the righ angles when two 3-4-5 right traingles are put together to form a cyclic quadrilateral.

* indicates level of diffuclty.