A Characterization of the Euler Line

Let $V_1V_2V_3$ be a triangle with circumcenter $O$, and let $M_1,M_2,M_3$ be the midpoints of $V_2V_3, V_1V_3$, and $V_1V_2$, respectively. 

For $−∞<t≤∞$ and $k=1,2,3,$ let Mtk be the point defined by $OMt_k = tOM_k$, (where by $Mt_∞$ we mean the point at infinity in the direction of $OM_k$.) 

Prove that for any $t∈(−∞,∞]$, the lines $V_kMt_k, k=1,2,3, $are concurrent, and that the locus of all such points of concurrence is the Euler Line of $ΔV_1V_2V_3$.

Read more »