By A. P. Kiselev, Alexander Givental
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Extra info for Kiselev's Geometry / Book II. Stereometry
When t he center of homothety lies in the plane of the polygon. T herefore this remains true for any center due to the lemma (since polygons congruent to similar ones are similar). (3) A polyhedron obtained from a given one by a homothety with a positive coefficient is similar to it. It is obvious that corresponding elements of homothetic polyhedra a re positioned similarly with respect to each other, a nd it follows from the previous two corollaries that the polyhedral angles of such polyhedra are respectively congruent a nd corresponding faces similar.
POLYHEDRA 34 Therefore C'A 2 = AB 2 + BC 2 + C'C 2 . Corollary . Jn a rectangular parallelepiped, all diagonals are congruent. 58. Parallel cross sections of pyramids. Theorem. pyramid (Figure 49) is intersected by a plane parallel to the base, then: (1) lateral edges and the altitude (SM) are divided by this plane into proportional parts; (2) the cross section itself is a polygon (A'B'C'D'E') similar to the base; (3) the areas of the cross section and the base are proportional to the squares of the distances from them to the vertex.
Theorem (tests for congruence of trihedral angles). Two trihedral angles are congruent if they have: (1) a pair of congruent dihedral angles enclosed between two respectively congruent and similarly positioned plane angles, or (2) a pair of congruent plane angles enclosed between two respectively congruent and similarly positioned dihedral angles. A B' ~igur e 38 (1) Let S and S' be two trihedral angles (Figure 38) such t hat L. A'S'C' (and these respectively congruent angles are also positioned similarly), and the dihedral angle AS is congruent to t he dihedral angle A'S'.