  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.
