Question 119
Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ in such a way that $P$ lies between $B$ and $Q$, and rays $AP$ and $AQ$ trisect the angle $BAC$. The line parallel to $AQ$ and passing through $P$ meets the side $AB$ of the triangle at point $D$, and the line parallel to $AP$ and passing through $Q$ meets the side $AC$ of the triangle at point $E$. Can it happen that $DE$ is a midsegment of the triangle $ABC$?