GeoProof — Euler Line explorer

O circumcenter G centroid H orthocenter N nine-point center

O, G, H are collinear · OG : GH = 1 : 2

The Euler Line

Every triangle has several famous "centers." Three of them — the circumcenter O, the centroid G, and the orthocenter H — are always lined up:

O, G, H collinear, with OG : GH = 1 : 2

The straight line through them is the Euler line. The centroid always sits one-third of the way from the circumcenter to the orthocenter.

Meet the centers

O is where the three perpendicular bisectors of the sides meet (center of the circumscribed circle). G is where the three medians meet (the balance point, the average of the vertices). H is where the three altitudes meet. Toggle each construction on to see how its center is found.

The nine-point center too

The center N of the nine-point circle is the midpoint of OH — so it also lies on the Euler line, exactly halfway between the circumcenter and orthocenter. Four centers, one line.

When the line disappears

For an equilateral triangle all these centers land on the same point, so there is no unique Euler line. Make the triangle nearly equilateral and the four points crowd together; stretch it and they spread out along the line.

A bit of history

Leonhard Euler proved the collinearity of O, G, and H in 1765 — one of many gems bearing his name.