Desargues’ theorem

Desargues’ theorem

Let

A

⁢

B

⁢

C

and

X

⁢

Y

⁢

Z

be two

triangles

. They are said to be

perspective

from a point

if

A

⁢

X

,

B

⁢

Y

and

C

⁢

Z

are either

concurrent

or

parallel

, and are said to be

perspective from a line

if either the points of

intersections

A

⁢

B

∩

X

⁢

Y

,

B

⁢

C

∩

Y

⁢

Z

and

C

⁢

A

∩

Z

⁢

X

all exist and are

collinear

, or do not exist at all (three pairs of

parallel lines

).

Given two triangles such that no vertex of one triangle is the vertex of another:

•

The

Desarguesian property

states: if they are perspective from a point, they are perspective from a line.

•

The

dual Desarguesian

property

states: if they are perspective from a line, they are perspective from a point.

A related

concept

is that of a

Desarguesian configuration

, which consists of two triangles which are both perspective from a point and perspective from a line. We say that the two triangles form a Desarguesian configuration. The point and the line are called the

vertex

and

axis

of the

configuration

. Note that the point may be a

point at infinity

, and the line may be a

line at infinity

. Below is a diagram of a Desarguesian configuration.

(XEukleides

\PMlinktofile

source codedesargues.euk for the drawing)

A

geometry

with points, lines and an

incidence relation

between them is said to be

Desarguesian

if, given any two triangles such that no vertex of one is the vertex of another, then both the Desarguesian property and its dual are true. Equivalently, a geometry is Desarguesian if whenever two triangles are in perspective from either a point or a line, then they form a Desarguesian configuration.

Desargues’

theorem

. The

Euclidean space

is Desarguesian.

Remarks

.

•

In general,

affine spaces

and

projective spaces

are Desarguesian, provided that the space in question is at least

dimension

3. If the dimension is 2, it can be shown that the space (affine or projective) is Desarguesian iff it can be embedded in a 3 dimensional space (affine or projective).

•

In order to show that a projective space is Desarguesian, one only needs to show one of the two Desarguesian properties, since the other one may be automatically deduced according to the principal of

duality

. In

addition

, one may drop the case where two lines are parallel, as they always intersect at a point. In proving Desargues’ theorem, one generally “

complete

” the affine space into a projective one first, and use

homogeneous coordinates

to prove the theorem.

•

A special type of Desarguesian configuration where its vertex lies on its axis is called a

minor Desarguesian configuration

. The

minor Desarguesian property

states that if two triangles are perspective from a point, then they form a minor Desarguesian configuration. Interchanging points and lines, we may form the dual minor Desarguesian property.

•

Another special type of Desarguesian configuration occurs when the geometry is affine. A Desarguesian configuration is said to be

affine

if its axis is a line at infinity. In other words, given two triangles perspective from a point, they form a Desarguesian configuration in which their corresponding sides are parallel. An affine Desarguesian configuration is minor if its vertex is a point at infinity. In other words, the two triangles are such that not only are their corresponding sides parallel, the lines joining the corresponding vertices are parallel as well. These statements may also be dualized.

Title

Desargues’ theorem

Canonical

name

DesarguesTheorem

Date of creation

2013-03-22 11:51:32

Last modified on

2013-03-22 11:51:32

Owner

drini (3)

Last modified by

drini (3)

Numerical id

19

Author

drini (3)

Entry type

Theorem

Classification

msc 51A30

Classification

msc 46L05

Defines

Desarguesian

Defines

dual Desarguesian

Defines

Desarguesian property

Defines

affine Desarguesian

Defines

minor Desarguesian

Defines

Desarguesian configuration

Defines

minor affine Desarguesian

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