For every $P$ point $Q$ is picture of $P$ at rotational homothety at $A$ with rotational angle $\pm45^{\circ}$ and $k= {\sqrt{2}\over 2}$. So $P$ describes lines $\ell '$ and $\ell ''$ which makes $45^{\circ}$ angle with $\ell$ and goes thrue $A'$ which is orthogonal projection of $A$ on $\ell$.