Question 10
In an acute triangle $ABC$, let $D$ be point on the side $[BC]$ different than the vertices. Let $M_1, M_2, M_3, M_4, M_5$ be the midpoints of the line segments $[AD], [AB], [AC], [BD], [CD]$, respectively; $O_1, O_2, O_3, O_4$ be the circumcenters of the triangles $ABD, ACD, M_1M_2M_4, M_1M_3M_5$, respectively; $S$ and $T$ be the midpoints of the line segments $AO_1$ and $AO_2$, respectively. Prove that $SO_3O_4T$ is an isosceles trapezoid.