Consider the (mostly nontrivial) tetrahedron formed by the three moving points and P. There is a determinantal formula for the volume of the tetrahedron, as well as one for the area of the triangle formed by the three moving points. Now you have two formulas depending on t, one measuring signed tetrahedral volume, the other the signed area of the moving triangle. If the area is nonzero at time t while the volume is 0 at time t, then you have a time when point P is in the plane of the triangle. Then you can do some calculations to see if P is inside the triangle or not. Otherwise, either the volume is nonzero (and thus no intersection between P and the triangle), or the triangle has zero area, and you have to check to see if P is collinear at that time with the other points, and also lies within the segment determined by the three points.
If you have more information about the trajectories, I may have more suggestions on how to find time t, if it exists.
Gerhard "Ask Me About System Design" Paseman, 2010.01.23
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