# 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?

### A challenge

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.