Euler line proof
Let the circumcenter of and its centroid. Extend until a point such that . We’ll prove that is the orthocenter
.
Draw the median where is the midpoint of . Triangles and are similar
, since , and . Then and . But so , that is, is a height of the triangle.
Repeating the same argument for the other medians proves that lies on the three heights and therefore it must be the orthocenter .
The ratio is since we constructed it that way.