Desargues assumption

Desargues theorem

If the corresponding sides of two triangles $ABC$ and $A'B'C'$ intersect at points $P,Q,R$ on the same straight line, then the straight lines which connect the corresponding vertices intersect at one point. Conversely, if the straight lines connecting the corresponding vertices of the triangles $ABC$ and $A'B'C'$ intersect at one point, the corresponding sides of these triangles intersect at points lying on a straight line. The converse statement for triangles lying in the same plane is dual to the statement for straight lines by the little duality principle. In both cases the points and the straight lines constitute a Desargues configuration (a $10_3$-configuration), located in some two-dimensional or three-dimensional projective space.

Figure: d031320a

If both triangles belong to the same projective space, the Desargues assumption cannot be demonstrated on the strength of plane incidence axioms alone (cf. Non-Desarguesian geometry), but it is valid for any projective plane which may be imbedded in a projective space of larger dimension. The spatial case of the Desargues assumption follows from the space incidence axioms.

The validity of the Desargues assumption is necessary and sufficient for the construction of the projective algebra of points of the projective straight line and for the synthetic introduction of projective coordinates. Established by G. Desargues, [1].

Comments

An affine or projective plane in which the Desargues assumption holds is called a Desarguesian plane.

In lattice-theoretical terms (cf. Lattice) the Desargues assumption may be formulated as the identity ([1])

$$[(x+z)(y+u)+(x+u)(y+z)](x+y)\leq$$

$$\leq[(y+x)(z+u)+(y+u)(z+x)](y+z)+$$

$$+[(z+y)(x+u)+(z+u)(x+y)](z+x).$$

References

[1]	A. Bosse, "Manière universelle de m-r Desargues, pour pratiquer la perspective par petit-pied comme le géométral" , Paris (1648)
[2]	D. Hilbert, "Grundlagen der Geometrie" , Springer (1913)
[3]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[a1]	E. Artin, "Geometric algebra" , Interscience (1957)
[a2]	D. Pedoe, "A course of geometry" , Cambridge Univ. Press (1970)
[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)

L.A. Skornyakov

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