Geometry: Theorems

Problem : In what kind of triangle is the orthocenter also the circumcircle?

Problem : In what kind of triangle is the centroid also the incircle?

Problem : What does it say about a triangle if one of its sides is also an altitude

Problem : What must be true of a triangle if its circumcenter lies in the interior of the triangle?

Problem : If segments AD and BF are medians of triangle ABC, how long is segment AD?