A plane is uniquely defined by 3 non-collinear points. The equation of the plane has a geometric interpretation based on areas and volumes.
Form a triangle using points A, B, and C. Project this triangle onto each coordinate plane, and compute the area of each of these projected triangles [call these areas Area xy, Area xz, and Area yz]. Next, compute the volume of the tetrahedron formed by A, B, and C and the origin [call this Vol].
The equation of the plane through A, B, and C is (Area yz) x + (Area xz) y + (Area xy) z = 3 Vol