1. 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.

2. solution to the first one:
Let D, E, F to be the mid-points of BC,CA and AB respectively. given that the medians AD,BE,CF are all equal.
The medians are concurrent at G.
now GD=GE=GF=(1/3)AB.
this shows that G is the circumcentre of triangle DEF.
also EF is parallel to BC. so by basic proportionality theorem AG bisects the side EF. Since G is the circumcentre, the line joining the mid-point of EF to G is perpendicular to EF. since EF is parallel to BC the line AG extented(which id the median AD) meets BC at right angles. So the median AD is perpendicular to BC. similarly the other two medians also meet the corresponding sides at right angles. this will easily lead to the conclusion that the triangle ABC is equilateral.

3. solution to the second one:
Let E and F be the feet of altitude from the point B and C respectively.
Since H is the orthocentre the lines BE and CF pass through H. now $\angle BEA+\angle CFA = 180$. Hence the quadrilateral AEHF is cyclic. Proved.

4. I cannot understand the third problem statement. please repost the problem in this thread again. thanks!

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

Two 3-4-5 right traingles are put together to form a cyclic quadrilateral. So the two triangles have side 3 , 4 , 5 . Find the distance between the vertices at the right angles.

I hope this clarified and omg thank you sooo sooo much for the help ! this helped me soo much! =]

6. from what i can understand the cyclic quadrilateral becomes a rectangle. now we have to find the diagonal's length which in this case is equal to 5.