Euler line

The previous post discussed the circumcenter and orthocenter of a triangle. Euler proved that the centroid, circumcenter, and orthocenter all fall on a common line, now called the Euler line.

The centroid is the center of mass of a triangle. If you draw lines from each vertex to the midpoint of the opposite side, the three lines intersect at the centroid.

The center of the nine-point circle also lies on the Euler line, though this was discovered after Euler.

Trilinear coordinates

Let α. β, and γ denote the three angles of the triangle.

The previous post said that the trilinear coordinates, of the circumcenter are

cos α : cos β : cos γ

and the trilinear coordinates of the orthocenter are

sec α : sec β : sec γ.

The trilinear coordinates of the centroid are

csc α : csc β : csc γ.

The center of the nine-point circle has trilinear coordinates

cos β−γ : cos γ−α : cos α−β.

Here’s a visualization for a particular triangle.

triangle with circumcenter, centroid, nine-point center, and orthocenter

The dashed line in the figure above is the Euler line for a particular triangle. The colored dots, moving from northwest to southeast, are the orthocenter, nine-point center, centroid, and circumcenter.

For an equilateral triangle, all four of these points coincide.

Proof outline

Discovering the Euler line required a flash of insight. Verifying it does not.

Three points are colinear if the 3×3 matrix formed by stacking their trilinear coordinates has zero determinant. So we could prove Euler’s original theorem by showing that the determinant of the matrix

\begin{bmatrix} \cos\alpha & \cos\beta & \cos \gamma \\ \sec\alpha & \sec\beta & \sec \gamma \\ \csc\alpha & \csc\beta & \csc \gamma \\ \end{bmatrix}

is zero. This isn’t zero for arbitrary α, β, and γ. But since these are angles in a triangle,

γ = π − α − β

and with that additional information we can show that the determinant is zero. We could replace one of the rows with the trilinear coordinates of the nine-point circle to show that it also lies on the Euler line.