Collection of Geometry Problems: Planimetry and Stereometry (Russian Math Classics — English Editions)

The mathematician who founded the Soviet olympiad movement wrote a geometry problem book. This is it, in English for the first time.

In 1934, Boris Delone and his student V. A. Tartakovskii organised the first school mathematics olympiad in the USSR. Three years later Delone published this collection with O. K. Zhitomirsky and A. I. Fetisov: 85 geometry problems, every one worked through in full. The 1941 edition is translated here complete.

Delone is the Delaunay of the Delaunay triangulation — the "empty sphere" construction that today underlies computational geometry, mesh generation, CAD and GIS. He spent the 1930s doing two things at once: inventing the mathematics, and teaching Soviet schoolchildren to think like mathematicians. This book is the second of those.

What makes this different from a modern problem book:

Every solution is a method, not an answer. Construction problems are solved in four movements — analysis, construction, proof, investigation. Analysis reasons backwards from the finished figure to find the idea. Construction carries it out. Proof shows it is correct. Investigation asks when a solution exists at all, and how many there are. Those four headings appear 111 times in this book. Most problem books print the answer and stop.

The investigation is the point. Problem 18 asks you to construct a quadrilateral from its four sides and the segment joining two midpoints. The solution does not stop at a construction: it works through eight possible correspondences of vertices, derives the condition for each, and settles exactly when the problem has no solution, one, two, or infinitely many. That is what a mathematician actually does with a problem.

It goes further than you expect. The Simson line. Menelaus's theorem, proved in both directions. The Euler nine-point circle. Homothety in the plane and in space. The rhombic dodecahedron, cut from a cube by twelve planes at 45 degrees, with a volume exactly twice the cube's. Every symmetry axis of all five regular solids. How to construct the diameter of a sphere using nothing but a compass laid on its surface. The volume common to a cube and a copy of itself rotated about an axis.

108 pages. 85 problems. 75 figures, each reproduced at the size it has in the original. Two chapters of method — one on construction problems, one on problems in space — then the problems themselves in planimetry (the straight line, the circle, areas) and stereometry (lines, planes and polyhedral angles; polyhedra; round bodies), then the complete solutions.

This is a faithful translation from the Russian — no adaptation, no simplification. Problem numbering, figures and the order of the solutions follow the source edition.

The book outlived one of its authors. O. K. Zhitomirsky died of starvation in early 1942, during the siege of Leningrad, when the city was encircled by German and Finnish troops.

Pair with the problem collections in the same tradition. Rybkin's Collection of Problems in Geometry, Part I: Planimetry and Part II: Stereometry are the classical Russian problem books, both now in first complete English editions.

Continue with Kiselev's Geometry, Book I: Planimetry and Book II: Stereometry — the exposition these problems assume, by the author whose textbooks taught Russian and Soviet schools for over seventy years.

Ideal for: students preparing for mathematical olympiads and competitions, teachers who want problems that carry a method rather than a trick, homeschool families looking for rigorous classical geometry, undergraduates rebuilding their geometric foundations, and anyone curious about the training that produced the Soviet mathematical school.

Part of Russian Math Classics — English Editions. Translated by Valery Manokhin, PhD (Royal Holloway, University of London). Published by Northern Star Academic Press. Full catalogue at russianmathbooks.com.