On a theorem about projectivities

Research paper by Charles Thas

Indexed on: 01 Nov '96Published on: 01 Nov '96Published in: Journal of Geometry


The main result of this paper is a theorem about projectivities in then-dimensional complex projective spacePn (n ≥2). Puttingn = 2, we showed in [3] that the theorem of Desargues inPn is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inPn. We only give a few examples and leave it as an open problem which other special cases could be found.