MathNet 2019-part-01 — 184 problems (EN–VI) Question 150

Let $AA_1$ be the bisector of the triangle $ABC$. The points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively.
Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$.
(M. Karpuk)