What is the relation between orthocenter, circumcentre, and centroid?

In geometry, the orthocenter, circumcenter, and centroid are three important points of concurrency in a triangle. Their relationship is beautifully explained through the concept of the Euler line.

The orthocenter (H) is the point where the three altitudes of a triangle intersect. The circumcenter (O) is the point of intersection of the perpendicular bisectors of the sides, and it is also the center of the circumscribed circle of the triangle. The centroid (G) is the point where the three medians of the triangle meet, and it represents the triangle’s center of mass.

In any non-equilateral triangle, these three points—H, O, and G—lie on a single straight line called the Euler line. The centroid divides the line joining the orthocenter and the circumcenter in a fixed ratio of 2:1, with the centroid always closer to the circumcenter. Symbolically:

HG = 2 × GO

This relation highlights the balance within a triangle: the centroid acts as a "bridge" between the orthocenter and circumcenter.

In the special case of an equilateral triangle, the orthocenter, centroid, and circumcenter coincide at the same point, making the Euler line degenerate into a single point.