# The Euler Segment of a Triangle

Drag the vertices P, Q, and R to adjust the triangle.

The eminent 18th century mathematician
Leonard Euler
discovered a remarkable relationship among the various "centers" of a triangle, illustrated here.

He showed that the centroid, B, always lies between the circumcenter, C, and orthocenter, O, on a line segment (shown in orange) which we now call the *Euler Segment*, in his honor.

### Things to think about and explore

When does the Euler segment lie entirely inside the triangle? When does it extend beyond the triangle?

What happens to the Euler segment when the triangle is somehow special, for example, when it is a right triangle or isosceles triangle?

When does the incenter lie on the Euler segment?

Notice that not only is the centroid always between the circumcenter and the orthocenter, it is half as far from the circumcenter as from the orthocenter. If you know about similar triangles and their properties, try to show that the point on the line joining the circumcenter and centroid, twice as far from the centroid as the circumcenter and on the opposite side *is* the othocenter.

If you're still wondering about what the orthocenter and Euler segment are good for, you might enjoy contemplating the Nine Point Circle.

If you have Geometer's Sketchpad, you might enjoy the sketch upon which this page is based.

### Some terminology

- The
*Centroid* or *Barycenter* of a triangle is the intersection of its medians, that is the lines joining its vertices to the midpoints of the opposite sides (shown here in purple). It is the center of gravity of a triangular plate of the same shape.
- The
*Circumcenter* of a triangle is the intersection of the perpendicular bisectors (shown in blue-green) of its sides. It is the center of the smallest circle that contains the triangle, the *Circumscribed Circle*.
- The
*Incenter* of a triangle is the intersection of its angle bisectors (not shown). It is the center of the largest circle contained in the triangle, the *Inscribed Circle*.
- The
*Orthocenter* of a triangle is the intersection of its altitudes, that is the perpendiculars from its vertices to the opposite sides (shown in red).
*Similar triangles* have the same angles, but not neccessarily the same side lengths. In other words they are the same, but for a scale factor.

Copyright © 2003 Joe Christy joe @ eshu.net.

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