Nine-point circle

Builds on Triangle Midpoint Circumcircle Altitude of a triangle Orthocentre

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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Altitude of a triangle concept
Circumcircle concept
Midpoint concept
Orthocentre concept
Triangle concept
Nine-point circle definition

Tag	`0J768YF1ZE`
URI	`m2://0J768YF1ZE`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Nine-point circle
Uses	Triangle, Midpoint, Circumcircle, Altitude of a triangle, Orthocentre

Nine-point circle

Definition

Remarks

Reading path

Further reading

What this page needs for W2

What links here