C.a.R. > Applications > Triangle Geometry > Euler line

• In the green triangle, U is the orthocenter, as we saw in the beginning. In the black one, U is the center of the circumcircle.
• Moreover, U* is the center of the circumcircle of the green triangle.
• S is on AA*, BB* and CC*. Consequently, it is the barycenter of the black triangle. Since the green triangle is the dilatation of the black one, S is also the barycenter of the green triangle.

Thus, S is between U and U* and SU=2SU*.

The green circle is the nine point circle of the black triangle.