Question 186
Let $ABC$ be an acute triangle and $F$ its Fermat point, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of triangles $ABF$, $BCF$ and $CAF$, draw its Euler line, that is, the line connecting its circumcenter and its centroid.
Prove that these three lines pass through one common point.
Prove that these three lines pass through one common point.