The lines $a$, $b$ and $c$ are given, point $A_1$ on line $a$ and point $B_1$ on line $b$. Conic $\mathcal{C}$ passes through points $A_1$ and $B_1$, and lines $a$, $b$ and $c$ are its tangents. Construct tangent $d$ of the conic which is parallel to $c$ (and different from $c$). In which cases there is no solution?
I applied Brianchon's theorem to the "hexagon" $aabbcc$ in order to find touching point $C_1$ of tangent $c$ and conic $\mathcal{C}$. However I don't know what to do next. How do I proceed from this?
