The above shows three steps toward a solution of this flash game inside a triangle.   Quite a mess at first, followed by one last node placement to sort out, followed by the planar graph solution.  Triangles don't make the best use of screen space and so make things more difficult in the higher levels of play.  Squares are shown below:
And finally, some pentagon closed-loop solutions.  Screen grabs from during this action-packed round of planarization: