#### Proof of Theorem 5

**Theorem 5**

Suppose that **C: **__x__^{T}M__x__=0 and
**D: **__x__^{T}N__x__=0 are non-degenerate conics.

Then the dual of **D** with respect to **C** is the non-degenerate conic with equation
__x__^{T}MN^{-1}M__x__=0.

**Proof
**

We think of **D** as the line-conic (the envelope of its tangents),
**{**__u__^{T}N__x__=0 : __u__^{T}N__u__=0}.

The dual of __u__^{T}N__x__=0 with respect to **C** is the point
**V=[**__v__], where __v__= M^{-1}(N__u__).

Then __u__= N^{-1}(M__v__).

Since **[**__u__] is on **D**, __u__^{T}N__u__=0, and so
__v__^{T}MN^{-1}NN^{-1}M__v__=0, *i.e.*
__v__^{T}MN^{-1}M__v__=0.

Thus the dual of **D** with respect to **C** has equation
__x__^{T}MN^{-1}M__x__=0, as required.