As told in the last post, we now come to the Problems in Plane Geometry by I. F. Sharygin in the Science for Everyone series.

In the preface Sharygin says:

This volume contains over 600 problems in plane geometry and consists of two parts. The first part contains rather simple problems to be solved in classes and at home. The second part also contains hints and detailed solutions. Over 200 new problems have been added to the 1982 edition, the simpler problems in the first addition having been eliminated, and a number of new sections- (circles and tangents, polygons, combinations of figures, etc.) having been introduced, The general structure of the book has been changed somewhat to
accord with the new, more detailed, classification of the problems. As a result, all the problems in this volume have been rearranged.

From the back cover of the book:

This is a translation from the revised edition of the Russian book which was issued in 1982. It is actually the first in a two-volume work on solving problems in geometry, the second volume “Problems in Solid Geometry” having been published in English first by Mir Publishers in 1986. Both volumes are designed for schoolchildren and
teachers.

This is a unique collection of interesting and elegant problems presenting the subject in a manner comprehensible to a youthful mind making it both interesting and usefulthrough a wide range of practical applications. Although the problems in this collection vary in “age” (some of them can be found in old books and journals, others were offered at mathematical olympiads or published in the journal “Quant” (Moscow, we still hope that some of the problems in the collection will be of interest to experienced geometers.

Almost every problem is non-standard (as compared with routine exercises on solving equations, inequalities, etc.): one has to think of what additional constructions must be made, or which formulas and theorems must be used. Therefore, this collection cannot be regarded as a problem hook in geometry; it is rather a collection of geometrical puzzles aimed at demonstrating the elegance of elementary geometrical techniques of proof and methods of computation (without using vector algebra and with a minimal use of the method of coordinates, geometrical transformations, though a somewhat wider use of trigonometry).

The book was translated from the Russian by Leonid Levant and was first published by Mir in 1988.

You can get the book here. Link updated 01/Feb/2017

Facing Problems: FAQs

The contents of the book are as under:

Contents
Preface to the English Edition . 6
Section 1 Fundamental Geometrical Facts and Theorems. Computational Problems  8

Section 2. Selected Problems and Theorems of Plane Geometry 65.
Carnot’s Theorem 65.
Ceva’s and Menelaus’ Theorems. Affine Problems 70.
Loci of Points 82.
Triangles. A Triangle and a Circle 89.
Quadrilaterals 117.
Circles and Tangents. Feuerbach’s Theorem 129.
Combinations of Figures. Displacements in the Plane. Polygons 137.
Geometrical Inequalities. Problems on Extrema 147

Answers, Hints, Solutions 160

Section 1 160
Section 2 .214

Appendix. Inversion 392