Fitting a triangle on 3 lines

We present a solution to the following fitting problem: Let $g_1$, $g_2$, $g_3$ be arbitrary lines in space and let $T_1^0 T_2^0 T_3^0$ be some triangle. Find poses $T_1 T_2 T_3$ of $T_1^0 T_2^0 T_3^0$ with $T_i \in g_i$, $i = 1, 2, 3$.
We also characterize the configurations of $g_1$, $g_2$, $g_3$ and $T_1 T_2 T_3$ admitting a motion where $T_i$ runs on $g_i$, $i = 1, 2, 3$.