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:

CentroidThe 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.

CircumcenterThe 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. SeeCircumcenter of a Triangle for more.

OrthocenterThe 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.

Nagel Line

The Nagel line is the term proposed for the first time in this work for the line on which the incenter  , triangle centroid , Spieker center Sp, and Nagel point Na lie. Because Kimberling centers   and   both lie on this line, it is denoted   and is the first line in Kimberling's enumeration of central lines containing at least three collinear centers (Kimberling 1998, p. 128).

The Kimberling centers   lying on the line include   (incenter  ), 2 (triangle centroid  ), 8 (Nagel point Na), 10 (Spieker center Sp), 42, 43, 78, 145, 200, 239, 306, 386, 387, 498, 499, 519, 551, 612, 614, 869, 899, 936, 938, 975, 976, 978, 995, 997, 1026, 1103, 1125, 1149, 1189, 1193, 1198, 1201, 1210, 1644, 1647, 1698, 1714, 1722, 1737, 1961, 1998, 1999, 2000, 2057, 2340, 2398, 2534, 2535, 2664, 2999, 3006, 3008, 3009, 3011, and 3017.

The Nagel line is central line  , so its trilinear equation is

(1)

The Nagel line satisfies the remarkable property of being its own complement, and therefore also its own anticomplement.

The incenter  , Spieker center Sp, Nagel point Na, and triangle centroid   satisfy the distance relations

(2)

(3)

The Nagel line is the radical line of the de Longchamps circle and Yff contact circle.