Triangles (6) ... The Euler Line

In any triangle, the  circumcentre, centroid, and orthocentre always lie on a straight line, called the Euler line. The line is named after Leonard Euler (1707 - 1783) and he showed that those points are collinear in 1765 

In equilateral triangles, the points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them.

The incentre generally does not lie on the Euler line and is on the Euler line only for isosceles triangles,  for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers. The incentre is the point where the bisectors of the interior angles intersect.

The diagram shows △ ABC and O is the circumcentre, G is the centroid, H is the orthocentre. Other points also lie on the Euler line - e.g. the De Longchamps point (L) and the Nine-Point Centre N. 

G always lies 1/3 of the way from O to H - in other words the ration OG:GH is 1:2
N always lies 1/2 of the way from O to H;
O always lies 1/2 of the way from H to L.  

The Centroid is the point where the three medians of the triangle intersect.

The Circumcentre is the point where the three perpendicular bisectors of the triangle meet.  

The Orthocentre is the point where the three altitudes of a triangle intersect.  

Proof that the points are collinear - Polymathematics - Euler's Line

GoGeometry - Euler Line of a Triangle