Existence of the Euler Line: An Elementary Proof

 
 
Vladimir Nikolin

In any triangle, the orthocenter, the centroid and the circumcenter are collinear on the Euler line. In addition HG = 2GO.


Figure 1

    Let D be the point opposite from C on the circumcircle, Mc be the midpoint of side AB. We are already proved that the quadrilateral ADBH is a parallelogram and, consequently, the point MC is midpoint of line segment HD, or

HM_C = M_CD

    Let's focus on ΔHCD. From above follows that CMc serves as a median in both triangles ABC and HCD , which means that they share the centroid G, too. But, HO is another median of ΔHCD and

G = CM_C \cap HO
so points H, G and O are collinear. The common point ,centroid G, of the two medians CMc and HO divides each in the ratio 2:1, therefore HG = 2GO.