The three harmonic homologies theorem

In this paper, we give a complete answer to the question: “Under what conditions the product of three harmonic homologies of the real projective space $P^n\left(R\right)$ is a harmonic homology again?” Among other things, we prove the three harmonic homologies theorem in $P^n\left(R\right)$ 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. It is shown 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.