The EllipseTangent Theorem If E is an ellipse and L is a line, then (1) the family <L> contains exactly two tangents to E, and (2) the tangents to E at P and Q are parallel if and only if PQ is a diameter. The proof depends on the corresponding result for the circle: The Circle
For any circle C, each family of parallel lines contains exactly two tangents


Proof By Theorem AC2, there is an affine transformation s mapping E to a circle C. As s is affine <L> maps to the parallel family <s(L)>. By the Circle Theorem, The latter family contains exactly two tangents M and M'. As s^{1} is also affine, s^{1}(M) and s^{1}(M') are the only two tangents in <L>.
Finally, since the centre C of C lies on the line joining the points of contact of 
