Lately I’ve been thinking a lot about Euler and his many contributions to mathematics.
So, just for fun I decided to make a Geogeba applet showing the “Euler Line.”
In 1763 Euler proved that three different centers of a triangle—the centroid, the orthocenter, and the circumcenter—are collinear. This line is called the Euler line.
The centroid of a triangle is the triangle’s center of mass. It is located at the point of intersection of the three medians of the triangle. Recall that a median joins a vertex to the midpoint of the opposite side.
The orthocenter of a triangle is the point of intersection of the three altitudes. An altitude is a line through a vertex, perpendicular to the opposite side.
Finally, the circumcenter is the point of intersection of the perpendicular bisectors of the three sides of the triangle. The circumcenter is also the center of the circle that circumscribes the triangle.
Incidentally, Euler also proved that the distance between the centroid and the orthocenter is twice the distance between the centroid and the circumcenter.
(Also on this line, but not shown in the applet, is the center of the nine-point circle, which happens to be the midpoint between that triangle’s orthocenter and circumcenter.)