Desargues' theorem

In projective geometry, Desargues' theorem, named in honor of Girard Desargues, states:

In a projective space, two triangles are in perspective axially if and only if they are in perspective centrally.

To understand this, denote the three vertices of one triangle by (lower-case) a, b, and c, and those of the other by (captial) A, B, and C. Axial perspectivity is the condition satisfied iff the point of intersection of ab with AB, and that of intersection of ac with AC, and that of intersection of bc with BC, are collinear, on a line called the axis of perspectivity. Central perspectivity is the condition satisfied iff the three lines Aa, Bb, and Cc are concurrent, at a point called the center of perspectivity.

In an affine space nothing similar is true unless one lists various exceptions involving accidentally parallel lines. Desargues' theorem is therefore one of the most basic of simple and intuitive geometric theorems whose natural home is in projective rather than affine space.

The truth of Desargues' theorem in the plane is most readily deduced by getting it as a corollary to its truth in 3-space. Two triangles cannot be in perspective unless they fit into a space of dimension 3 or less; thus in higher dimensions the span of the two triangles is always a subspace of dimension no higher than 3.

The ten lines involved (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. (It is an amusing exercise to show that those incidence conditions can also be satisfied by a configuration of ten points and ten lines that is not incidence-isomorphic to the Desargues configuration.) The statement of the theorem above may misleadingly connote that the Desargues configuration has less symmtery than it really has: 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 perspecitivity.