 Solution

Let the triangles have vertices $A_0,A_1,A_2$ and $B_0,B_1,B_2$. Work initially with nonzero-area triangles. A triangle-triangle intersection algorithm begins with the unnormalized normal vector $N_A=(A_1-A_0)\times (A_2-A_0)$ and the three signed plane evaluations $d_j=N_A\cdot (B_j-A_0)$. If all $d_j$ are strictly positive, or all strictly negative, the second triangle lies strictly on one side of the first one's plane, so the triangles are disjoint. Otherwise compute $N_B$ and $e_i=N_B\cdot (A_i-B_0)$ and apply the symmetric rejection test. Strict inequalities are important: contact on a vertex or edge counts as intersection.

If the planes are not parallel, put $L=N_A\times N_B$. Each triangle meets the other one's plane in an empty set, a point or a line segment. Its endpoints are obtained from zero-valued vertices and the edges whose endpoint plane evaluations have opposite signs. For example, an edge $B_j{B}_k$ contributesAll these points lie on the common line of the two planes. Choose a coordinate $r$ for which $L_r\ne 0$ and form the coordinate intervals $I_A=[a_{-},a_{+}]$, $I_B=[b_{-},b_{+}]$ of the two slices. This coordinate is one-to-one along the common line, so the exact final test isOnly that coordinate of each slice endpoint needs to be evaluated; no full three-dimensional line parameterization is necessary.

Parallel planes which have survived the strict plane tests are coplanar. Drop the coordinate corresponding to the largest component of $N_A$ and test the two projected planar triangles. For each edge of either projected triangle, project both vertex sets onto the perpendicular direction. A strictly disjoint pair of projection intervals certifies separation. If none of the six edge directions separates them, the planar convex sets overlap: a separating line for two disjoint convex polygons can be chosen parallel to an edge of one of them. This includes containment and boundary contact.

A zero normal vector signifies a degenerate triangle; reduce its convex hull to a point or its extreme line segment and use point containment or segment intersection instead. For floating-point input, reliable sign predicates and scale-aware error bounds are preferable to treating a tiny signed distance as an arbitrary exact zero. Plane rejection followed by interval overlap, with a planar test for coplanar input, determines all nondegenerate cases.