** 4.4 Desargues' Theorem
***Printout*

*If Desargues, the daring pioneer of the
seventeenth century, could have foreseen what his ingenious method of projection
was to lead to, he might well have been astonished. He knew that he had done
something good, but he probably had no conception of just how good it was to
prove.
—*

The French mathematician Gérard
Desargues (1593–1662) was one of the earliest contributors to the study of
synthetic projective geometry. Desargues was an engineer and architect, who had
served in the French army. The importance of the theorem, that bears his name, is
due to the relating of two aspects of projective geometry: perspectivity from a
point and perspectivity from a line. Because of his many contributions to the
field of projective geometry, the theorem was named after him even though his
major work,* Brouillon projet*, was lost for nearly two centuries before
another French geometer Michel Chasles (1793–1880) discovered a copy in 1845.

Though we are only studying plane projective geometry, we motivate Desargues'
Theorem with a triangular pyramid in three dimensions. The diagram on the left
is a triangular pyramid with vertices *A, B, C,* and* P*. The triangles *ABC*
and *A'B'C'* are perspective from the point *P*. From Euclidean
geometry, two nonparallel planes intersect in a line. Therefore, the two planes*
a *and* a'
*determined by the triangles* ABC* and *A'B'C'* intersect in a line*
l*. Since* AB* and* A'B'* are in planes* a
*and* a'*,
respectively, the point* Q* =* AB · ^{ }A'B'* must be on line

*Axiom 5. (Desargues' Theorem) If two triangles are perspective
from a point, then they are perspective from a line.*

Click here for a dynamic illustration of Desargues' Theorem GeoGebra or JavaSketchpad.

In plane projective geometry,
Desargues' Theorem cannot be proven from the other axioms; therefore, it is
taken as an axiom. The proof of the theorem requires two triangles that are not
in the same plane, as illustrated in the motivation example above. That is, Desargues' Theorem can be proven from the other
axioms only in a projective geometry of more than two dimensions. Since we have
not listed the axioms for a projective geometry in 3-space, we will not discuss
the proof of the theorem here, but the proof is similar to the argument made in
the illustration above. Many books on projective geometry discuss the
topic. (Reference *Projective Geometry* by
Veblen and Young*, *1938)* *

*Dual of Desargues' Theorem. If two triangles
are perspective from a line, then they are perspective from a point.*

** Exercise 4.16. **Construct two triangles that
are perspective from a point, then determine the line from which the triangles
are perspective.

** Exercise 4.17. **Construct two triangles that
are perspective from a line, then determine the point from which the triangles
are perspective.

** Exercise 4.18.** Use dynamic geometry software
for this exercise. In the Poincaré
Half-plane, construct two triangles that are perspective from a point.
Investigate whether the triangles are also perspective from a line.

** Exercise 4.19. **Given
Δ

4.3 Principal of Duality in Projective Geometry4.5.1 Harmonic Sets |

© Copyright 2005, 2006 - Timothy Peil |