Nine-point circle tangent to the incircle and the radius of the nine-point circle is half the radius of the circumcircle.
Nine-point circle goes through:
1. the midpoint of each side of the triangle
2. the foot of each altitude (the point where the altitude touches the side to which it is perpendicular)
3. the midpoints between the orthocenter (point where altitudes meet) and each vertex
The incircle is tangent to each side of the triangle. The center of the incircle is the concurrence of all the angle bisectors of the triangle.
The circumcircle goes through each vertex of the triangle. The center of the circumcircle is the concurrence of the perpendicular bisectors of each side of the triangle.

Drag one of the vertices of the triangle around to change its shape. When do the two of the circles coincide or disappear?
Find the point of tangency between the nine-point circle and the incircle.
Prove that the radius of the circumcircle is twice the radius of the nine-point circle.