Lecture Notes 7: Desargues Theorem

Not following the text, we will first state this result for projective planes and then specialize it to affine planes. The theorem is named for the French mathematician Girard Desargues (1593-1662) who proved it in Euclidean geometry. In more general geometries, it need not always be true. Thus, it is not really a theorem, just a property that may or may not hold, but it is traditional to refer to the statement as a theorem. Those projective (and affine) planes in which it is true are known as Desarguesian projective (affine) planes. An example of an affine plane which is not a Desarguesian affine plane is the Moulton Plane given in the text.

In a projective plane, two triangles are said to be perspective from a point if the three lines joining corresponding vertices of the triangles meet at a common point called the center. Two triangles are said to be perspective from a line if the three points of intersection of corresponding lines all lie on a common line, called the axis. Desargues' Theorem: If two triangles are perspective from a point then they are perspective from a line.

A triangle is a self-dual object. The dual statement of "two triangles are perspective from a point" is "two triangles are perspective from a line" and vice versa. Thus, the dual of Desargues theorem is the converse of that statement, namely, "if two triangles are perspective from a line, then they are perspective from a point." Even though the converse is the dual statement, one can not prove the converse by applying the principle of duality (as the text implies, but does not state). The reason is that the principle of duality only applies to those statements which are true for all projective planes and the Desargues theorem is not such a statement. The principle of duality could be used if it can be shown that Desarguesian planes are self-dual (which is true), but the proof of that statement is as complicated as the direct proof of the converse, so nothing is gained by taking this approach.

There is no necessary relationship between the center and axis of a pair of triangles. In the older literature, the special case occurring when the center was on the axis, Desargues theorem was known as the "Little" (or minor) Theorem of Desargues (but this is no longer fashionable).

We will not, at this time, provide the conditions that are necessary and sufficient for a projective plane to be a Desarguesian projective plane. Rather, we shall assume that this property holds and consider the consequences of this assumption.

Affine Versions of Desargues Theorem

In the event that the axis of a pair of perspective triangles is the "line at infinity" of an affine plane obtained by removing the "line at infinity" from a Desarguesian projective plane, the conclusion of the Desargues theorem has to be rephrased in terms of parallels. There are two cases to consider. The general situation is given by:

Affine Desargues Theorem (Case II): If two triangles in an affine Desarguesian plane are perspective from a point and two pairs of corresponding sides are parallel, then the third pair of corresponding sides are parallel.

The other special case corresponds to the little theorem of Desargues. Here, we can not even refer to two triangles being perspective from a point, since the center is a point at infinity and is not in the affine plane. The statement is:

Affine Desargues Theorem (Case I): If the lines joining pairs of corresponding vertices of two triangles are mutually parallel and two pairs of corresponding sides are parallel, then the third pair of corresponding sides of the triangles are parallel.

There are other special cases of Desargues theorem in affine planes (obtained by letting some elements of the basic configuration be "at infinity"), but these two are all that are needed for our purposes.