 Polygonal connectivity of a finite-line complement

The complement of finitely many lines in ${\mathbb{R}}^3$ is path-connected, and any two points can be joined by two line segments. For endpoints $a,b$ outside the lines, exclude the finitely many affine planes through each endpoint and each removed line. Choose a bend point $c$ outside all those planes. If either segment met a removed line, the bend point would lie in its associated excluded plane, a contradiction.