Proof of the Common Tangents Theorem

The Common Tangents Theorem

Suppose that C and D are distinct conics.
Then the common tangents (if any) to D and C
occur at the points where C intersects the dual of D with respect to C.

Throughout the proof duality means duality with respect to C.

Let D' be the dual of D. Then, of course, D is the dual of D'.

Suppose first that D' meets C at T.
As T is on C, the dual of T is the tangent to C at T.
As D is the dual of D', the dual of T is a tangent to D,
so is a common tangent to C and D.

Suppose now that there is a common tangent L to C and D, meeting C at T, say.
As L is the tangent to C at T, T is the dual of L.
As L is also a tangent to D, T, the dual of L, must lie on D', the dual of D.
Thus T lies on the intersection of C and the dual of D.

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