Euler's Discovery

The circumcenter is the point where all the perpendicular bisectors of the sides of a triangle intersect. It can be used to create a circle around the triangle. The orthocenter is where the lines that are perpendicular to the sides and pass through the opposite vertex meet. The Centroid is the point where the lines that pass through the midpoints of each side and the opposite vertex.
What was the circumcenter good for? What was the orthocenter good for? What was the centroid good for? What was the discovery? Why might this happen?