6. Given a circle and its center O, to draw a line through a given point P parallel to a given line q. Prove the following construction: Let p be the polar of P, Q the pole of q, and A the intersection of p with OQ. The polar of A is the desired line.
CHAPTER VIII - INVOLUTION
Fig. 32
121. Fundamental theorem. The important theorem concerning two complete quadrangles (§ 26), upon which the theory of four harmonic points was based, can easily be extended to the case where the four lines KL, K'L', MN, M'N' do not all meet in the same point A, and the more general theorem that results may also be made the basis of a theory no less important, which has to do with six points on a line. The theorem is as follows:
Given two complete quadrangles, K, L, M, N and K', L', M', N', so related that KL and K'L' meet in A, MN and M'N' in A', KN and K'N' in B, LM and L'M' in B', LN and L'N' in C, and KM and K'M' in C', then, if A, A', B, B', and C are in a straight line, the point C' also lies on that straight line.
The theorem follows from Desargues's theorem (Fig. 32). It is seen that KK', LL', MM', NN' all [pg 72] meet in a point, and thus, from the same theorem, applied to the triangles KLM and K'L'M', the point C' is on the same line with A and B'. As in the simpler case, it is seen that there is an indefinite number of quadrangles which may be drawn, two sides of which go through A and A', two through B and B', and one through C. The sixth side must then go through C'. Therefore,