Is there a good method of Identifying a triangle that is utterly redundant on a polygon?

I'm trying to write a function for C++ that can examine a union of two polygons and eliminate any leftover faces that didn't get correctly culled because of floating point error or various other small-number glitchy reasons.

Is there a good method by which one can identify a 3D triangle that is completely coplanar and contained within other faces on a 3D polygon and thus is completely redundant?

It's easy enough to detect a triangle within another TRIANGLE, but I'm looking for a situation where I can detect a redundant triangle that might cross and be contained several triangles that form a flat plane.

For example, in the picture above, assuming all vertices are in one plane, the red triangle is completely unnecessary and will only result in z-fighting. And yet I can't seem to come up with a good technique that covers all bases for saying "the red one can be removed."

Any suggestions?

Posting the solution I came up with here in case this question ever can help anyone with a similar problem. What I did was:

1. Identify all triangles lying in the same plane
2. Rotate those into 2D triangles along that plane (this step is to reduce floating point precision problems and reduce the math needed in the next step)
3. If any given triangle has ALL THREE POINTS within other triangles in this list it is redundant and can be removed.