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.
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.
26. Fundamental theorem concerning two complete quadrangles. This theorem throws into our hands the following fundamental theorem concerning two complete quadrangles, a complete quadrangle being defined as the figure obtained by joining any four given points by straight lines in the six possible ways.
Given two complete quadrangles, K, L, M, N and K', L', M', N', so related that KL, K'L', MN, M'N' all meet in a point A; LM, L'M', NK, N'K' all meet in a [pg 17] point Q; and LN, L'N' meet in a point B on the line AC; then the lines KM and K'M' also meet in a point D on the line AC.
