projective.




*25. Desargues’s theorem.* We consider first the following beautiful
theorem, due to Desargues and called by his name.

_If two triangles, __A__, __B__, __C__ and __A’__, __B’__, __C’__, are so
situated that the lines __AA’__, __BB’__, and __CC’__ all meet in a point,
then the pairs of sides __AB__ and __A’B’__, __BC__ and __B’C’__, __CA__
and __C’A’__ all meet on a straight line, and conversely._

[Figure 3]

FIG. 3


Let the lines _AA’_, _BB’_, and _CC’_ meet in the point _M_ (Fig. 3).
Conceive of the figure as in space, so that _M_ is the vertex of a
trihedral angle of which the given triangles are plane sections. The lines
_AB_ and _A’B’_ are in the same plane and must meet when produced, their
point of intersection being clearly a point in the plane of each triangle
and therefore in the line of intersection of these two planes. Call this
point _P_. By similar reasoning the point _Q_ of intersection of the lines
_BC_ and _B’C’_ must lie on this same line as well as the point _R_ of
intersection of _CA_ and _C’A’_. Therefore the points _P_, _Q_, and _R_
all lie on the same line _m_. If now we consider the figure a plane
figure, the points _P_, _Q_, and _R_ still all lie on a straight line,
which proves the theorem. The converse is established in the same manner.