Incidentally, I don't know how to do this, I'm just trying to figure it out.
Now, given three points that define a plane, (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), and having determined that (z3 != z1 or z2 != z1) {the plane does intersect the x - y axial plane}, with our purpose therefore being to derive the equation for the intersection (a line) of the plane and the x - y axial plane, our next step would be ...
But, first, three points define a plane and also three lines. Calling the three points P1, P2, and P3, those are a line through P1 and P2, which we can call L1, a line through P1 and P3 which we can call L2, and a line through P2 and P3 which we can call L3.
Having established this, let us assert that if the plane (which we have elsewhere called P) intersects the x - y axial plane, then at least two of the three lines, L1, L2, and L3, must also intersect the x - y plane, and at two different points. But, there's a catch. It is possible that all three of the lines do intersect the x - y plane (and they would do so at three different points, all three of which would be on a line, that is, on the intersection of P and x - y), but it is also possible that one of the lines L1, L2, and L3 does not intersect x - y. Well, we only need two points to derive the P - x - y intercept, which we can call P - i, but we need to choose the correct two lines, out of L1, L2, and L3, in the case that one of them is parallel to Pxy (naming the x - y plane thus).
If (z1 = z2) {L1 does not intersect}.
If (z1 = z3) {L2 does not intersect}.
If (z2 = z3) {L3 does not intersect}.
If (P interesects P - i and L1 does not intersect P - i) {
Pi1 is the intersection of L2 and P - i and Pi2 is the intersection of L3 and P - i}, and, applying the same test for L2 and L3, find Pi1 and Pi2. If L1, L2, and L3 all intersect P - i, we can derive Pi1 and Pi2 using two of them arbitrarily.
OK, that's a little silly.
If (z1 != z2) {find Pi1 using L1}, and continue with L2, or, if L2 does not intersect, L3.
Next, given a line that does intersect, defined by points (xa, ya, za) and (xb, yb, zb), how do we find the point Pi where that line intersects P - i (where Pi is the intersection of the line, thus, a point, with the x - y axial plane, P - i)?
