Coxeter- Theorem 6.42

Author:
LexiPasi
Topic:
Geometry
6.42 A quadrangle and a quadrilateral, with the four vertices of the former associated in a definite order with the four sides of the latter, are related by just one projective correlation Let defpqr and D’E’F’P’Q’R’ be the given quadrangle and quadrilateral. Suppose A is x.y with x through d.e and y through d.q. The projectivities between def and D’E’F’, and between dqr and D’Q’R’ determine a line a’=X’Y’, where (defx is projectively related to D’E’F’X’) and (dqry is projectively related to D’Q’R’Y’). Let A vary in range, so that x is perspective with y. By our construction for a’, we now have X’ is projectively related to x, which is perspective with y, which is projectively related with Y’. Since d is an invariant line of the perspectivity between x and y, D’ must be an invariant point of the projectivity between X’ and Y’. Thus a’ varies in pencil. The projectivity between x and X’ suffices to make it a projective correlation, because we have a point-to-line transformation and line-to-point transformation preserving incidence. Finally, there is no other projective correlation transforming defpqr into E’E’F’P’Q’R’; for if another transformed A into a1, the inverse of the latter would take a1 to A, the original correlation takes A to a’, and altogether, we would have a projective collineation leaving D’E’F’P’Q’R’ invariant and taking a1 to a’. Because the only transformation that satisfies this is the identity, the correlation is unique.