The three harmonic homologies theorem (Q6197582)

The authors give a complete answer to the question: ``Under what conditions the product of three harmonic homologies of the real projective space \(P^n(\mathbb{R})\) is a harmonic homology again?'' Among other things, they prove the three harmonic homologies theorem in \(P^n(\mathbb{R})\) by which the product of three harmonic homologies with collinear centers is again a harmonic homology if and only if the hyperplanes are polars of the centers with respect to a quadric. They show that the three reflections theorem, the three inversions theorem, notably Pascal's theorem and Miquel's theorem in Laguerre geometry are special cases of this theorem.