Russian geometry book

Recently I received the following note,

I would like to bring to your attention the following new
geometry textbook:

"Kiselev's Geometry / Book I. Planimetry" by A.P. Kiselev,
ISBN 0977985202 Publisher: Sumizdat

It is an English translation and adaptation of a classical Russian textbook in plane geometry, which has served well as to several generations of students of age 13 and up, and their teachers in Russia.

The English edition is intended for those students, homeschooled or not, who want to achieve a good command of elementary geometry, and learn to appreciate for its intellectual depth and beauty.

More information about the book and its author is available through the publisher's webpage:

The book is currently available at: and

I posted this note here because some of you might be interested - a classical Russian geometry book translated into English. You can browse quite many sample pages to get an idea of the book.
It was first published in 1892 has been revised and published more than forty times altogether.

The original author Kiselev wrote several math textbooks. QUOTING from the preface:

"...and a few years prior to Kiselev's death in 1940, his books were officially given the status of stable, i.e. main and only textbooks to be used in all schools to teach all teenagers in Soviet Union.

The books held this status until 1955 when they got replaced in this capacity by less successful clones written by more Soviet authors. Yet "Planimetry" remained the favorite under-the-desk choice of many teachers and a must for honors geometry students. In the last decade, Kiselev's "Geometry," which has long become a rarity, was reprinted by several major publishing houses in Moscow and St.- Petersburg in both versions: for teachers as an authentic pedagogical heritage, and for students as a textbook tailored to fit the currently active school curricula. In the post-Soviet educational market, Kiselev's "Geometry" continues to compete successfully with its own grandchildren.

Quite an accomplishment for a single book.

If you look at the type of exercises found in the book, I think you will easily see that there is a difference when comparing to modern American books (this book is meant for 7-9th graders).

For example (these are from the sample pages provided on the website)

79. Suppose that an angle, its bisector, and one side of this angle in one triangle are respectively congruent to an angle, its bisector, and one side of this angle in another triangle. Prove that such triangles are congruent.

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

Prove theorems:

400. If a diagonal divides a trapezoid into two similar triangles, then this diagonal is the geometric mean between the bases.

401. If two disks are tangent externally, then the segment of an external common tangent between the tangency points is the geometric mean between the diameters of the disks.

402. If a square is inscribed into a right triangle in such a way that one side of the square lies on the hypotenuse, then this side is the geometric mean between the two remaining segments of the hypotenuse.

576. The altitude dropped to the hypotenuse divides a given right triangle into smaller triangles whose radii of the inscribed circles are 6 and 8 cm. Compute the radius of the inscribed circle of the given triangle.

577. Compute the sides of a right triangle given the radii of its circumscribed and inscribed circle.

578. Compute the area of a right triangle if the foot of the altitude dropped to the hypotenuse of length c divides it in the extreme and mean ratio.

Personally, I feel they are interesting sounding problems! (I will probably solve some in future blogposts, as examples.)

But how many US high school students would be willing and able to do them? Feel free to comment.

Now, this book could serve for a high school geometry course for sure. It does have one big disadvantage though if you're a homeschooler: there is no answer key. But the book appears to possess intellectual depth and beauty, just like its subject matter!

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Anonymous said…
US students, I don't know, but I'm a Norwegian high school student interested in it :)
Though I'm pretty jealous at my Russian classmate's planimetry book in Russian (Zadachi po planimetrii). Looks aweseome, and though there is an English translation in the public domain, it just isn't as cool as a thick, problem-rich textbook.
Anonymous said…
I find problem 79 difficult to understand. That is, I don't understand what figure being described should look like.

Recently I have been reading Heath's translation of Euclid's Elements and the greatest difficulty for me has been to understand the convoluted language of the translation. This can be a large obstacle to understanding. There is a two fold task: to think geometrically, and to speak and read geometrically.

An anonymouse ;)
Unizor said…
I'd like to introduce to you an educational Web site UNIZOR.COM dedicated to teaching advanced mathematics to teenagers. The Geometry part of this Web site is very much influenced by this Kiselev's book. The site is free and I invite everybody to study using it. It has a lot of problems (including many from the Kiselev's book) with solutions and as exams. I'd like to also mention that the goal of this Web site is not to provide mathematical facts, but to develop student's creativity, logical thinking and analytical approach to any, not only math, problems.

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