Hello

,

a random point $\displaystyle Q$ was chosen on side $\displaystyle AC$ of triangle $\displaystyle ABC$. Point $\displaystyle P$ is the center of side $\displaystyle BC$. Segments $\displaystyle AP$ and $\displaystyle BQ$ meet at $\displaystyle T$. Point $\displaystyle R$ is the center of $\displaystyle AT$, while point $\displaystyle S$ belongs to $\displaystyle BT$ and $\displaystyle BS=QT$. Show that $\displaystyle PS||QR$.

It's gotten too complicated for me. I'm hoping for your kind help.