Why Are the Circumcenter, Centroid, and Orthocenter Collinear?

The theorem

Theorem

The Euler line

Let $O$ , $G$ , and $H$ be the circumcenter, centroid, and orthocenter of a non-equilateral triangle $ABC$ . Then $O,G,H$ are collinear and

OG:GH=1:2.

Their common line is the Euler line of $ABC$ .

The circumcenter as an orthocenter

Let $D,E,F$ be the midpoints of $BC,CA,AB$ , respectively. By the midpoint theorem,

EF\parallel BC,\qquad FD\parallel CA,\qquad DE\parallel AB.

The perpendiculars to $BC,CA,AB$ through $D,E,F$ are the three perpendicular bisectors of $ABC$ , and hence meet at $O$ . By the parallel relations above, these same lines are the three altitudes of $DEF$ . Thus $O$ is the orthocenter of the medial triangle.

Triangle ABC with medial triangle DEF; the three altitudes of DEF meet at O, the circumcenter of ABC. — Figure 1. The circumcenter O of ABC is the orthocenter of the medial triangle DEF.

The homothety and the Euler line

Since $G$ divides each median in the ratio $2:1$ , the homothety centered at $G$ with ratio $-2$ sends

D\mapsto A,\qquad E\mapsto B,\qquad F\mapsto C.

It therefore carries $DEF$ onto $ABC$ . Since homotheties preserve perpendicularity, it sends the orthocenter $O$ of $DEF$ to the orthocenter $H$ of $ABC$ . Hence

\overrightarrow{GH} =-2\overrightarrow{GO}.

So $O,G,H$ are collinear and

OG:GH=1:2.

Triangle ABC and its medial triangle DEF, with centroid G and the homothety of ratio minus two sending O to H on the Euler line. — Figure 2. The homothety centered at G with ratio -2 sends D, E, F, O to A, B, C, H, respectively.

The nine-point center

Let $N$ be the circumcenter of $DEF$ . The same homothety sends $N$ to $O$ , while it sends $O$ to $H$ :

N\mapsto O\mapsto H.

Since its ratio is $-2$ ,

GO=2GN,\qquad GH=4GN,

with $N$ and $H$ on one side of $G$ and $O$ on the other. Hence

ON=NH=3GN.

Thus $N$ is the midpoint of $OH$ . Since the circumcircle of $DEF$ is the nine-point circle of $ABC$ , $N$ is the nine-point center.

The Centroid through Ceva and Vectors
Two Views of the Orthocenter
The Nine-Point Circle from the Circumcircle shows how a homothety centered at $H$ produces all nine points at once.
Diagram and construction source

The theorem

The circumcenter as an orthocenter

The homothety and the Euler line

The nine-point center

Related notes and diagrams