#### Proof of Theorem 2

**Theorem 2**

Suppose that **C: **__x__^{T}M__x__=0 is a non-degenerate conic and
that **VW** is a chord of **C** passing through a fixed point **U**.

Then the tangents at **V** and **W** meet on the algebraic polar of **U** with respect to **C**.

**Proof**

By Theorem 1, the tangents at **V=[**__v__] and **W=[**__w__] have equations
__v__^{T}M__x__=0 and
__w__^{T}M__x__=0, respectively.

Suppose that these meet in **T=[**__t__]. By definition, **T** is on the
geometrical polar of **U**.

As **VW** passes through **U=[**__u__], __u__ = a__v__ + b__w__,
for some **a**, **b**.

Then
__u__^{T}M__t__=a__v__^{T}M__t__+__w__^{T}M__t__=0
since **T** lies on each tangent.

Thus **T** lies on __u__^{T}M__x__=0,the
algebraic polar of **U**.