**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.*

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

HM_C = M_CD

Let's focus on ΔHCD. From above follows that CM_{c} 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 CM_{c}and HO divides each in the ratio 2:1, therefore HG = 2GO.