Feuerbach's and Poncelet's theorems meet in space

Abstract

We propose 3D generalizations of the Feuerbach theorem: the first one deals with a tetrahedron analogue of the Euler circle, the second one is done by means of an {\guillemotleft}up-in-ex-touch{\guillemotright} construction. Then we give a geometric proof of the Grace theorem (a classical, but still not well-known, 3D Feuerbach theorem) and show its relation to the Poncelet closure theorem. Our elementary proof is based on the properties of imaginary generators on a sphere and of isotropic tangents to a conic. Then we show that the Grace theorem implies the Laguerre theorem, which generalises the Euler-Chapple formulas. Further, we consider a spatial analog of Poncelet's theorem. It is also proved that the Grace spheres touch a fixed sphere under the Poncelet rotation of bicentric tetrahedron. Finally, the lifting to the three-dimensional space provides a new proof of Feuerbach's theorem and, perhaps, the shortest proof of Euler-Chapple formulas.