Euler Line
Drag the three vertices of a triangle to watch four classical centers — circumcenter, centroid, orthocenter, and nine-point center — fall into perfect collinearity along the Euler line. Toggle each element to study the geometry.
About this lab
In 1765, Leonhard Euler proved that three of the most important centers of any triangle — the circumcenter (center of the circumscribed circle), the centroid (intersection of medians), and the orthocenter (intersection of altitudes) — are always collinear. The line passing through them is now called the Euler line. Moreover, the centroid always divides the segment from the circumcenter to the orthocenter in the ratio 1:2.
The nine-point center, discovered later by Feuerbach, is the midpoint of the segment from the circumcenter to the orthocenter, and it too lies on the Euler line. The nine-point circle passes through nine significant points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from the orthocenter to each vertex. Its radius is exactly half the circumradius.
There is one beautiful exception: for an equilateral triangle, all four centers coincide at a single point, and the Euler line is undefined. As you drag the vertices toward an equilateral configuration, you can watch the four centers converge. This collinearity is not a coincidence but a deep consequence of the affine and metric geometry of the Euclidean plane.