We say that

We take

It is easy to see that a circumconic for **T*** has an equation of the
form pyz+qzx+rxy=0, which is non-degenerate

provided that pqr is non-zero.

Definition

If AA',BB' and CC' meet in a point P, then we
say that the triangles ΔABC and ΔA'B'C' are* in perspective from* P.

We
refer to P as the *perspector* of the triangles.

Desargue's Theorem

If triangles **T** = ΔABC and
**T'** = ΔA'B'C' are in perspective from a point, then the intersections of
AB,A'B', of BC,B'C'

and of CA,C'A' are collinear.

We refer to the line of intersections as the *perspectrix* of the
triangles.

Theorem 1

If **C** is a circumconic of triangle
**T** = ΔABC, then the tangents to **C** at A,B,C from a triangle in
perpsective with **T**.

If **T** = **T*** and **C**:
pyz+qzx+rxy=0, then the perspector is [p,q,r], and the perspectrix is
x/p+y/q+r/z=0.

We give a proof later.

In general, we refer to the perspector of **T** and its tangential
triangle as the **C(T)**-perspector. For **T***, we refer

to it simply
as the **C**-perspector.

We often use duality with respect to the fixed conic **D**:
x^{2}+y^{2}+z^{2} = 0. Although this has no real points,
it is

non-degenerate as a complex conic.

Duality with respect to
**D** associates the point [u,v,w] with the line ux+vy+wz=0. Note that if the
point is real,

so is the line. Also, the reference points X,Y,Z correspond
to the lines x=0, y=0, z=0, the reference

triangle** T*** is
self-dual.

Since **D** has matrix I, earlier results show that, if a conic
**C** has matrix M, then its **D**-dual has matrix M^{-1}.

Again, if **C** is real, so is its dual.

As a first application we note that the **D**-dual of an inconic of
**T** is a circumconic of the dual triangle, and the

contact triangle for
**T** corresponds to the tangential triangle for the dual. Theorem 1 shows
that the latter

pair are in perspective. Duality shows that the perspectrix
corresponds to a perspector for **T** and its contact

triangle. Thus we
have

Theorem 2

If triangle **T** has inconic **I**,
then **T** and the triangle with vertices at the contacts of **T** with
**C** are in perspective.

As before, we talk of the **I(T)-** and **I**-perspectors. We can get
results for coordinates from the rest of Theorem 1.

The reference triangle
**T*** is self-dual, so if the inconic **I** corresponds to the
circumconic **C** : pyz+qzx+rxy = 0,

then the **I**-perspector is
[1/p,1/q,1/r]. Using the matrix for **C**, we get an equation for **I**
namely

We summarise some results on poles and polars of conics as follows
:

Suppose that a conic **C** has matrix M. Then

(1) the polar of the
point U= [**u**] = [u,v,w] is the line **x**^{T}M**u** =
0,

(2) the pole of the line **L** : ux+vy+wz = 0 is the point
M^{-1}**u**.

If **C** is the circumconic pyz+qzx+rxy = 0 and U = [u,v,w], then (1)
allows us to write the tangent to **C** at U as

the tangents directly as

Proof of Theorem 1

For the reference triangle and the
given conic, the tangents **Y***, **Z*** meet at X** = [-p,q,r].

The
line XX** has equation ry-qz = 0. This passes through P = [p,q,r]. By algebraic
symmetry, P is also on

the lines YY** and ZZ**. The general case follows
since we can apply a projective transformation to map the

vertices of any
triangle to X,Y,Z.