The * centroid* of a triangle is
the common intersection of the three medians. A median of a triangle
is the segment from a vertex to the midpoint of the opposite side.
Point D is the centroid of triangle ABC.

The * orthocenter* of a triangle
is the common intersection of the three lines containing the altitudes.
Below, point H is the orthocenter of triangle EFG.

The * circumcenter* of a triangle
is the point in the plane that is equidistant from the three vertices
of the triangle. The circumcenter is on the perpendicular bisector
of each side of the triangle. Point C is the circumcenter of the
triangle below:

The * incenter* of a triangle is
the point on the interior of the triangle that is equidistant
from the three sides. The incenter lies on the angle bisector
of each angle. Point I is the incenter of the triangle below:

The Nine Point Circle for any triangle passes through the three midpoints of the sides (green points), the three feet of the altitudes (red points), and the three midpoints of the segments from the respective vertices to the orthocenter (blue points).

The following is the Nine Point Circle for the triangle above, with center N: