Let $P(i,j):\mathbb{R}^2\to\mathbb{R}^2$ denote the projection of line $t_i$ onto line $t_j$, and let $T:\mathbb{R}^2\to \mathbb{R}^2=P(1,2)\circ P(2,3)\circ \dotsb \circ P(k-1,k)\circ P(k,1)$ (in other words, $T$ is the just the map that projects $t_1$ onto $t_2$ onto $t_3$ and so on, back to $t_1$ again). If not all lines are parallel, $T$ is a contraction, and has a (unique) fixed point by the Banach fixed point theorem, and we can just generate all the $P_i$ from this point. (If all lines are parallel, the problem is trivial.)