Problems and Solutions in Euclidean Geometry (Dover Books on Mathematics) by M. N. Aref (PDF)

Description

Based on classical principles, this book is intended for a second course in Euclidean geometry and can be used as a refresher. Each chapter covers a different aspect of Euclidean geometry, lists relevant theorems and corollaries, and states and proves many propositions. Includes more than 200 problems, hints, and solutions. 1968 edition.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This little book continues to amaze me. Where did the authors get their incredible treasure of exercises? There must be close to 800 of them in the book. As befits a book geared towards “a second course in Euclidean geometry”, most of them will be too hard even for honors geometry high school students (unless the teacher is very selective and provides copious hints), but by and large that’s not because they require a lot of advanced, specialized, technical knowledge. (“Advanced” results like Ceva and Menelaus, coaxal systems, etc., don’t even appear until after the first four chapters!) It’s just that they’re challenging in how they force you to search for ways to apply what you know to get the job done. (Anyone who has spent time wrestling with the harder exercises in a high school geometry textbook will know what I mean here.)Each chapter begins with a list of well-known theorems presumed to be familiar to the student, and then launches into a series of “Solved Problems”, each of which states some (usually unfamiliar!) result and then gives the proof in detail. The proofs are all completely “elementary”, using only things a student of high school geometry would recognize, and utterly synthetic (no coordinates!). Some of these proofs, especially ones that use clever moves with area (akin to Euclid’s proof of the Pythagorean theorem), or ingenious geometric constructions, are quite lovely. But I must admit that in some cases they seem overly complicated (and somewhat incomplete in the way they overlook issues of “betweenness and separation”, which is so common in a synthetic proof driven by the accidents of the way one happens to draw a figure conforming to the given hypotheses). This is mainly the price the authors pay for keeping everything “elementary” (and coordinate-free). A knowledgeable reader will sometimes feel the need to supply his or her own proof (say using coordinate methods or a bit of affine linear algebra), or the desire to shout out “but this is all obvious if we just apply a bit of projective geometry”.In closing, in my opinion it’s the book’s awesome set of exercises that is its main value. If you think you thoroughly understand the content of a high school geometry course, and have a good bit of “mathematical maturity”, try your hand at any of the exercises in the first four chapters.

⭐Great shape as described.

⭐I bought this 1968 book by Aref and Wernick at a local book store in 2011. The title speaks for itself. But I think no more books of this kind will be written again because traditional “straight-edge and compass” geometry largely stopped being taught in high schools around about 1970. I was very lucky to have an older teacher (Mrs. Dodwell) for my year-8 Euclidean geometry. It was the end of the road. I later found that Euclidean geometry had been relocated to 3rd year University optional courses.Since I had benefited from the mental training which the traditional ruler/compass geometry gives you, I naturally asked some Uni lecturers responsible for high school curricula why it had been dropped. The answer was that the high school teachers had complained that it was so difficult that they couldn’t teach it any more. There were just not enough teachers who could do the geometry themselves even. And at Uni, it had become an optional 3rd year extension subject for the more talented and enthusiastic students.In the 5th century BC, the study of Euclidean-style geometry (obviously not called “Euclidean” at that time!) was expected for anyone who wanted to be a general or a politician. This was not because it was directly useful, but because it was a training for the mind. A bit like doing intelligence tests all day every day to train the mind. The exercises in this book are an excellent “training for the mind” for all subjects in later life, including sciences and humanities, law and medicine, engineering and carpentry. It’s a huge pity that this mental training is no longer available in the high schools.The “basic” books for Euclidean geometry are readily available.”

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⭐”And Euclid is famously the second most printed book in history, so they say. But the geometry which was regarded as “elementary” in 300 BC is now beyond even mathematics graduates. Even discovering proofs for theorems in the 13 books of “the Elements” is a brain-twisting, brain-curdling exercise for modern mathematicians, although they were considered to be only a first course in 300 BC.Luckily this set of problems by Aref and Wernick does start off very gently with the easiest problems, and full solutions are given for the “Solved Problems”, which gives some idea of the level of detail expected for the proofs. But I think that the old-style Euclidean geometry has had its day. It has had 2000 years of popularity, which is extremely impressive, but I think the modern mind is no longer capable of the ascetic discipline required to immerse oneself in a style of geometry which is characterised more by the limitations of its scope than by the broad range of its applicability. The main limitation is, of course, using only ruler and compass, a somewhat arbitrary constraint.Therefore I do not think there will be another book like this. The original 1968 publication date coincides with the end of the road for old-school Euclidean geometry. It still remains, however, probably the best mental training exercise in problem-solving skills that was ever invented. Anyone who wants to learn how to think logically should exercise their mind on proofs of Euclidean geometry theorems.By the way, I notice that this book has nothing on conic sections, although it has a little on cones. So really, this book could be thought of as an intermediate level geometry problem book. The 3rd century BC had even more to offer than this!

⭐Good !

⭐Challenging book with many exercises.

⭐Perfect geometry book for Olympiad

⭐Nice one

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