There's a nice simple algorithm to check if a point is inside a polygon: the principle is this: draw a line from it to any point outside the polygon (for example with an X that is larger than any X for all the polygon vertices), and count how many edges it crosses. Iff this number is even, the point is outside of the polygon.
There are problematic cases where your test line goes straight through a vertex. You could think up an elaborate advanced scheme, taking the direction of the crossing into account, and counting such a crossing as +0.5 or -0.5 depending on its direction; or you could simply try again with a different line.
Combine this with tests on whether any edges from the polygons actually cross, and the basis is set.