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the ellipse tangent theorem

The Ellipse-Tangent 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
to C, and these occur at diametrically opposite points of C.

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
M and M', the centre s^-1(C) of E lies on the line joining the points of contact
of s^-1(M) and s^-1(M').

affine tangents page

The Ellipse-Tangent 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 to C, and these occur at diametrically opposite points of C.
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 M and M', the centre s^-1(C) of E lies on the line joining the points of contact of s^-1(M) and s^-1(M').