Euler line
 
In any triangle, the centroid, circumcenter and orthocenter always lie on a straight line, called the Euler line.
Try this Drag any orange dot on a vertex of the triangle. The three dots representing the three centers will always lie on the green Euler line.

In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the many centers of a triangle are always collinear, that is, they always lie on a straight line. This line has come to be named after him - the Euler line. (His name is pronounced the German way - "oiler"). The three centers that have this surprising property are the triangle's centroid , circumcenter and orthocenter.

In the figure above (press 'reset' first if necessary) the centroid is the black middle point on the line. The circumcenter is the magenta point on the left, and the orthocenter is the red point on the right. As you drag any of the triangle's vertices around, you can see that these points remain collinear, all lying on the green Euler line.

The three centers involved each have their own page describing them, but here is a brief overview:

Centroid

The centroid is the point where the three medians converge. In the figure above click on "show details of Centroid". The medians (here colored black) are the lines joining a vertex to the midpoint of the opposite side. See Centroid of a Triangle for more.

Circumcenter

The circumcenter is the point where the perpendicular bisectors of the triangle's sides converge. In the figure above click on "Show details of Circumcenter". The three perpendicular bisectors (here colored magenta) are the lines that cross each side of the triangle at right angles exactly at their midpoint. See Circumcenter of a Triangle for more.

Orthocenter

The orthocenter is the point where the three altitudes of the triangle converge. In the figure above click on "Show details of Orthocenter". The three altitudes (here colored red) are the lines that pass through a vertex and are perpendicular to the opposite side. See Orthocenter of a Triangle for more.

Equilateral Triangles

Another interesting fact is that in an equilateral triangle, where all three sides have the same length, all three centers are in the same place. In the figure above, adjust the vertices to try and get all three centers to come together. You will see that the triangle is equilateral.

Related triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises