We are still on the Science for Everyone series. This post is first of two books by I. F. Sharygin. The two books are problem and solution books in geometry. In this post the Problems in Solid Geometry is taken up. The next post would be Problems in Plane Geometry.
This book contains 340 problems in solid geometry and is a natural continuation of Problems in Plane Geometry, Nauka, Moscow, 1982. It is therefore possible to confine myself here to those points where this book differs from the first. The problems in this collection are grouped into (1) computational problems and (2) problems on proof.
The simplest problems in Section 1 only have answers, others, have brief hints, and the most difficult, have detailed hints and worked solutions. There are two reservations. Firstly, in most cases only the general outline of the solution is given, a number of details being suggested for the reader to consider. Secondly, although the suggested solutions are valid, they are not patterns (models) to be used in examinations. Sections 2-4 contain various geometric facts and theorems, problems on maximum and minimum (some of the problems in this part could have been put in Section 1), and problems on loci. Some questions pertaining to the geometry of tetrahedron, spherical geometry, and so forth are also considered here.
As to the techniques for solving all these problems, I have to state that I prefer analytical computational methods to those associated with plane geometry. Some of the difficult problems in solid geometry will require a high level of concentration from the reader, and an ability to carry out some rather complicated work.
The book was translated from the Russian by Leonid Levant and was first published by Mir in 1986.
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Contents are as under:
Section 1. Computational Problems 7
Section 2. Problems on Proof 37
Section 3. Problems on Extrema. Geometric
Section 4. Loci of Points 54
An Arbitrary Tetrahedron 59.
An Equifaced Tetrahedron 61.
An Orthocentric Tetrahedron 64.
An Arbitrary Polyhedron.
The Sphere 65.
An Outlet into Space 68.
Answers, Hints, Solutions 69