Take any triangle — any shape, any size. From it you can construct nine particular points, drawn from three separate geometric processes, and remarkably, all nine lie on a single circle. The result was proved by the German mathematician Karl Wilhelm Feuerbach in 1822, and the circle now bears his theorem’s name: the nine-point circle.

Three colors mark three constructions. The blue dots are the midpoints of the sides. The coral dots are the feet of the altitudes — where each altitude meets the opposite side. The teal dots are the midpoints between each vertex and the orthocenter, the point where the three altitudes meet. As the triangle morphs through three different shapes, the gold circle adjusts to pass through all nine points at every moment.

The size of the nine-point circle depends on the triangle. Its radius is exactly half the triangle’s circumradius: \(R_9 = R/2\). Its center sits at the midpoint of the segment connecting the orthocenter and the circumcenter — a point known as the nine-point center, and one of the four classical centers of a triangle that lie together on the Euler line.