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Journal for Geometry and Graphics 25 (2021), No. 1, 079--095
Copyright Heldermann Verlag 2021

From M. C. Escher's Hexagonal Tiling to the Kiepert Hyperbola

Federico Giudiceandrea
Maurits Srl, Via della Mendola 21, 39100 Bolzano, Italy

Luigi Grasselli
Dept. of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Reggio Emilia, Italy

Generalizing Fermat and Napoleon points of a triangle, we introduce the notion of complementary Jacobi points, showing their collinearity with the circumcenter of the given triangle. The coincidence of the associated perspective lines for complementary Jacobi points is also proved, together with the orthogonality of this line with the one joining the circumcenter and the Jacobi points. Involutions on the Kiepert hyperbola naturally arise, allowing a geometric insight on the relationship between Jacobi points, their associated perspective lines and Kiepert conics of a triangle.

Keywords: Triangle geometry, Escher, Napoleon Point, Fermat Point, Kiepert hyperbola.

MSC: 51M04; 00A66.

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