Proof of Theorem 5

Theorem 5
Suppose that C: x^TMx=0 and D: x^TNx=0 are non-degenerate conics.
Then the dual of D with respect to C is the non-degenerate conic with equation x^TMN^-1Mx=0.

Proof
We think of D as the line-conic (the envelope of its tangents), {u^TNx=0 : u^TNu=0}.
The dual of u^TNx=0 with respect to C is the point V=[v], where v= M^-1(Nu).
Then u= N^-1(Mv).
Since [u] is on D, u^TNu=0, and so v^TMN^-1NN^-1Mv=0, i.e. v^TMN^-1Mv=0.
Thus the dual of D with respect to C has equation x^TMN^-1Mx=0, as required.

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