finding intersections in 3D

I need to know how to find where two triangles intersect in 3D space; if they do, I need to split them. in other words: you have triangle ABC, and triangle PQR, and they are intersecting. They are both split into 2 parts, leaving 4 polygons. That means that I need to know where on the borders they meet, so that I can create a few new points there to describe a NEW figure. After all this spiltting, the remaining figures are then drawn in order by their distance from the camera.

Also, how can you tell which polygon is infront of the other? it IS possible to have the front polygon further away due to varying sizes and stuff
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