ProjectiveCollinearityQuadrangle

Projective Collinearity of a quadrangle

I borrow the term from Bogomolny (see his site: "Cut-The-Knot"). This is a projectivity defined through a quadrangle q = ABCD. Namely it is the uniquely defined projectivity that interchanges opposite vertices. In the figure below it is denoted by T and defined through its properties: T(A) = C, T(C) = A, T(B) = D, T(D) = B. Such a map is involutive (T^2 = I) and has as fixed points the set consisting of the diagonal e = EF of the quadrilateral together with the isolated point G, which is the intersection point of its interior diagonals. The point T(X) lies on the line GX and is such that together with the intersection point X', of GX with e, it forms a harmonic tetrad (X,T(X),G,X') = -1. T maps a side of the quadrangle to its opposite side and has also the following property, useful in various instances:
Consider four points I, K, H, J on the sides of the quadrangle and their images I* = T(I), ...etc. Joining the resulting points with lines one defines the quadrangle q* = LML*M*. It has indeed the property that opposite vertices are respectively images of one another through T. i.e. L* = T(L) , L = T(L*) etc. ...

The proof for L, L* follows from the fact that line IK, maps via T to I*K* and line HJ to H*J*. Thus, their intersection, which is L, maps to the corresponding intersection, which is L*. A similar argument shows also the collinearity of M, G and M*.

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