Desargues's theorem

Desargues never published this theorem, but it appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective (Manière universelle de M. Desargues pour practiquer la perspective) to a practical book on the use of perspective published in 1648.[1] by his friend and pupil Abraham Bosse (1602–1676).[2]

The importance of Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring.

In an affine space such as the Euclidean plane a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space.

By definition, two triangles are perspective if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be similar.

Under the standard duality of plane projective geometry (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual: axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration (below) is a self-dual configuration.[3]

This self-duality in the statement is due to the usual modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective triangles is axially perspective" and the dual of this statement was called the converse of Desargues's theorem and was always referred to by that name.[4]

Desargues's theorem holds for projective space of any dimension over any field or division ring, and also holds for abstract projective spaces of dimension at least 3. In dimension 2 the planes for which it holds are called Desarguesian planes and are the same as the planes that can be given coordinates over a division ring. There are also many non-Desarguesian planes where Desargues's theorem does not hold.

Pappus's hexagon theorem states that, if a hexagon AbCaBc is drawn in such a way that vertices a, b and c lie on a line and vertices A, B and C lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called Pappian. Hessenberg (1905)[7] showed that Desargues's theorem can be deduced from three applications of Pappus's theorem.[8]

The converse of this result is not true, that is, not all Desarguesian planes are Pappian. Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be commutative. A plane defined over a non-commutative division ring (a division ring that is not a field) would therefore be Desarguesian but not Pappian. However, due to Wedderburn's little theorem, which states that all finite division rings are fields, all finite Desarguesian planes are Pappian. There is no known completely geometric proof of this fact, although Bamberg & Penttila (2015) give a proof that uses only "elementary" algebraic facts (rather than the full strength of Wedderburn's little theorem).

The Desargues configuration viewed as a pair of mutually inscribed pentagons: each pentagon vertex lies on the line through one of the sides of the other pentagon.

The ten lines involved in Desargues's theorem (six sides of triangles, the three lines Aa, Bb and Cc, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ten points, and each of the ten points lies on three of the ten lines. Those ten points and ten lines make up the Desargues configuration, an example of a projective configuration. Although Desargues's theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.

This restricted version states that if two triangles are perspective from a point on a given line, and two pairs of corresponding sides also meet on this line, then the third pair of corresponding sides meet on the line as well. Thus, it is the specialization of Desargues's Theorem to only the cases in which the center of perspectivity lies on the axis of perspectivity.

A Moufang plane is a projective plane in which the little Desargues theorem is valid for every line.

• Pascal's theorem

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• Bamberg, John; Penttila, Tim (2015), "Completing Segre's proof of Wedderburn's little theorem", Bulletin of the London Mathematical Society, 47 (3): 483–492, doi:10.1112/blms/bdv021, S2CID 123036578
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• Desargues Theorem at MathWorld
• Desargues's Theorem at cut-the-knot
• Monge via Desargues at cut-the-knot
• Proof of Desargues's theorem at PlanetMath
• Desargues's Theorem at Dynamic Geometry Sketches