Vlados021 wrote: Let $AA_1$ be the bisector of a triangle $ABC$. 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 $l$. Vary $B_1$ linearly on $AB$. Therefore $C_1$ also varies linearly, and this gives the degrees of lines $B_1D$ and $C_1F$ is $1$. For $B_1 = B$, $B_1D=C_1F$, so degree of their intersection is also $1$, so locus is line. For $B_1 = A$ we get that this line passes through $A$. Now choose $B_1$ such that $AB_1 = AC_1$, then $B_1DC_1F$ is a parallelogram, so the locus is line parallel to $B_1D||C_1F$ through $A$.

But $D$ and $F$ are fixed and line $l$ varies.

DDIT on $B_1C_1ED$ WRT $A$ and projecting on $BC$ does the job.