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

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.