Midpoints and perpendiculars seem to belong to different stories until the circle reveals the common plot.

The nine-point circle is a circle associated with every nondegenerate triangle and passes through nine important points. These are the midpoints of the triangle’s three sides, the feet of its three altitudes, and the midpoints between each vertex and the orthocenter. Its center lies on the Euler line, halfway between the circumcenter and the orthocenter, and its radius is half the radius of the triangle’s circumcircle. The circle reveals a striking connection among several otherwise separate constructions within triangle geometry and is closely related to the Euler line and orthocenter. It is used in Euclidean geometry, triangle geometry, and geometric proofs.