The Feuerbach Circle of a Triangle

Consider the triangle ABC below (colored magenta). Several interesting points and lines can be constructed, and they all move as you drag about a vertex A, B, or C of the triangle. (By the way, you can place your mouse cursor over the diagram and press the return key to lift the diagram off the page. You can move and resize the window that appears. In fact, on some browsers you must resize the window.)

The centroid Z of the triangle

Along the sides of the triangle, you see the midpoints labelled A', B', and C'. The midpoint of the side BC is A', the midpoint of the side CA is B', and the midpoint of the side AB is C'. A line connecting a vertex of the triangle to the midpoint of the opposite side is called a median of the triangle. The medians of this triangle are AA', BB', CC', and they're colored green. Notice that they all meet at one point Z in the triangle, also colored green. This point is called the centroid of the triangle. Other names for the centroid are the barycenter and the center of gravity of the triangle. If you make a real triangle out of cardboard, you can balance the triangle at this point.

It can be shown that the centroid trisects the medians, that is to say, the distance from a vertex to the centroid Z is twice the distance from the centroid to the opposite side of the triangle. So, for instance, AZ is twice A'Z.

Incidentally, you can drag around other points besides the vertices A, B, and C. If you drag any other point, the figure is designed to swirl around the center N of the Feuerbach circle. An exception is N itself, and if you move it, the figure will slide along with it. Only moving A, B, or C will actually change the shape of the triangle.

The altitudes and the orthocenter H

There's yet another interesting "center" of the triangle, the orthocenter. An altitude of the triangle is a line drawn through a vertex perpendicular to the side of the triangle opposite the vertex. There are three altitudes: one is AD perpendicular to the side BC, the second is BE perpendicular to the side CA, and the third is CF perpendicular to the side AB. They're colored blue here. Note that when the triangle is obtuse, two of the altitudes lie outside the triangle, so they actually connect a vertex to the opposite side extended. In the case of a right triangle, two of the altitudes are actually sides of the triangle.

The altitudes of a triangle meet at a point, called the orthocenter, denoted here by H. For an acute triangle, the orthocenter lies inside the triangle; for an obtuse triangle, it lies outside the triangle; and for a right triangle, it coincides with the vertex at the right angle.

For fun, see what points and lines coincide for special triangles: isosceles triangles, right triangles, equilateral triangles, and right isosceles triangles.

The center N of the Feuerbach circle

Let Q, R, S be the midpoints of AH, BH, CH respectively. The point N divides HZ in the ratio one to three and it is the center of the Feuerbach circle which passes through A', B', C', D, E, F, Q, R, S. So the Feuerbach circle bisects AH, BH, CH. As it goes through nine special points it is also known as the nine point circle. The lines A'Q, B'R, C'S have been drawn in black. They all meet at N.

The Euler line HNZ of the triangle

These three "centers" of the triangle lie on one straight line, called the Euler line. ("Euler" is pronounced something like "Oiler" in English.) Leonhard Euler (1707-1783) was a very prolific mathematician known for his discoveries in many branches of mathematics ranging from number theory to analysis to geometry. For more about the Euler line, see the Euler Line of a Triangle page.

This page is shamelessly similar to the Euler Line of a Triangle page.
The text is similar and it utilizes the same Geometry Applet developed by
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610