For a triangle ABC, with sides of length a,b,c respectively, you might be able to solve by letting A lie at the origin of the plane, and B lie in the x-axis. Then the distance of each corner from the point will represent the radii, , , and , of three spheres with centres at A, B, and C respectively.
The intersection of all three spheres will be the point, p, which might be obtained from finding , which satisfies the following three equations for spheres simultaneously:
You can calculate the point , knowing that A is at the origin and also knowing the lengths a, b, and c.
The intersection of the first two spheres should give you the equation for a circle parrallel to the z-y plane. The x value that satisfies both this, and the equation of your third sphere will be your x ordinate.
Assuming the distance from the triangle to the point is not zero (the point does not lie in the plane) there should be two solutions - one for the point above the plane, and one for the point below. The modulus of either of these (z-ordinates) is the distance.