7 - The Nine-Point Circle

Summary

One of the most remarkable discoveries in nineteenth century Euclidean geometry is that there is one circle that contains nine significant points associated with a triangle. In 1765 Euler proved that the midpoints of the sides and the feet of the altitudes of a triangle lie on a single circle. In other words, the medial and orthic triangles share the same circumcircle. Furthermore, the center of the common circumcircle lies on the Euler line of the original triangle. More than fifty years later, in 1820, Charles-Julien Brianchon (1783–1864) and Jean-Victor Poncelet (1788–1867) proved that the midpoints of the segments joining the orthocenter to the vertices lie on the same circle. As a result, the circle became known as the “nine-point circle.” Later, Karl Wilhelm Feuerbach (1800–1834) proved that the ninepoint circle has the additional property that it is tangent to all four of the equicircles; for this reason Feuerbach's name is often associated with the nine-point circle.

The nine-point circle

Let us begin with a statement of the theorem.

Nine-point Circle Theorem. If ΔABC is a triangle, then the midpoints of the sides of ΔABC, the feet of the altitudes of ΔABC, and the midpoints of the segments joining the orthocenter of ΔABC to the three vertices of ΔABC all lie on a single circle.