#### Proof of Theorem 1

**Theorem 1**

If **C: **__x__^{T}M__x__=0 is a non-degenerate conic and
**U** is any point on **C**,

then the algebraic polar of **U** with respect to **C** is the tangent to **C** at **U
**.

**Proof**

Let **U=[**__u__], so that the algebraic polar is **L: **__u__^{T}M__x__=0.

Suppose that **L** cuts **C** again at **V=[**__v__].

As **U** lies on **C**, __u__^{T}M__u__=0.

As **V** lies on **C**, __v__^{T}M__v__=0.

As **V** lies on **L**, __u__^{T}M__v__=0.

Transposing this, and using the fact that **M** is symmetric, __v__^{T}M__u__=0.

Then, by expanding the left hand side, we see that

for any real **a **and **b**, **(a**__u__+b__v__)^{T}M(a__u__+b__v__)=0.

Thus **W=[a**__u__+b__v__] lies on **C**.

If **U** and **V** are distinct, then **C** contains the collinear points **U**, **V** and **W**,

but then **C** would be degenerate.

Thus **U=V**, so that **L** meets **C** just once, and hence is a tangent.