Tangency to two ellipses how?

I have arbitrary ellipse with center at <x,y>, major and minor axis <a,b> parallel to x and y axis, respectively, and a point on it P <c+x,d+y> at arbitrary angle T where c=a*cos(T) and d=b*sin(T).  If the major axis is a, the angle of tangency is atan2(d,a/cos(T)-c), and if the major axis is b, the angle is atan2(b/cos(90-T)-d,c).

The standard form of the tangent is y=tan(T)*x+(d+y)-tan(T)*(c+x).

I have two ellipses with arbitrary loci and axes.  How do I with similar simplicity determine the four lines of tangency those ellipses share?

Any two ellipses where one ellipse is not wholly contained by the other share at least two tangents.  If they intersect, or if they touch at a single point, they share two tangents.  If one is completely outside of the other, they share four tangents.