Question 169
A triangle $AB\Gamma$ is given and let $O$ be its circumcenter and $A_1, B_1, \Gamma_1$ the midpoints of its sides $B\Gamma, A\Gamma, AB$, respectively. We consider the points $A_2, B_2, \Gamma_2$ such that $\overrightarrow{OA_2} = \lambda \cdot \overrightarrow{OA_1}$, $\overrightarrow{OB_2} = \lambda \cdot \overrightarrow{OB_1}$ and $\overrightarrow{O\Gamma_2} = \lambda \cdot \overrightarrow{O\Gamma_1}$, with $\lambda \gt 0$. Prove that the lines $AA_2, BB_2, \Gamma\Gamma_2$ are concurrent.