Nine-point circle

Definition

Let $ABC$ be a triangle in the Euclidean plane $\mathbb{R}^2$. Let $M_A$, $M_B$ and $M_C$ be the midpoints of the sides $BC$, $CA$ and $AB$. These three points are not collinear. The nine-point circle of $ABC$ is the circumcircle of the triangle $M_A M_B M_C$.

Remarks

The circle also contains the feet of the three altitudes, and the midpoints of the segments $AH$, $BH$ and $CH$, where $H$ is the orthocentre. This gives the nine points of the name. Its centre is the midpoint of $H$ and the circumcentre $O$ of $ABC$. Its radius is half of the circumradius of $ABC$. Brianchon and Poncelet proved in 1821 that all nine points lie on one circle. In 1822, Feuerbach proved that the circle touches the incircle and the three excircles.

Explanations

L1 human

Join the midpoints of the sides of a triangle to get a smaller triangle. The circle through its corners is the nine-point circle. It also passes through six other special points of the big triangle.