TSTST 2022 Problem 2

Let $A B C$ be a triangle. Let $\theta$ be a fixed angle for which $$\theta<\frac{1}{2} \min (\angle A, \angle B, \angle C)$$ Points $S_{A}$ and $T_{A}$ lie on segment $B C$ such that $\angle B A S_{A}=\angle T_{A} A C=\theta$. Let $P_{A}$ and $Q_{A}$ be the feet from $B$ and $C$ to $\overline{A S_{A}}$ and $\overline{A T_{A}}$ respectively. Then $\ell_{A}$ is defined as the perpendicular bisector of $\overline{P_{A} Q_{A}}$. Define $\ell_{B}$ and $\ell_{C}$ analogously by repeating this construction two more times (using the same value of $\theta$ ). Prove that $\ell_{A}, \ell_{B}$, and $\ell_{C}$ are concurrent or all parallel.