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 is a harmonic homology again?” Among other things, we prove the three harmonic homologies theorem in 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.

中文翻译：

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三调和同调定理

在本文中，我们对“实射影空间的三个调和同调的乘积在什么条件下又是调和同调？”这个问题给出了完整的答案。除此之外，我们证明了三调和同调定理，其中三调和同调与共线中心的乘积再次是调和同调，当且仅当超平面是中心相对于二次曲面的极坐标。结果表明，拉盖尔几何中的三反射定理、三反转定理，特别是帕斯卡定理和米克尔定理是该定理的特例。