# Plane Geometry

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1. Sep 16, 2015

### Mastermind01

Hello,

I am totally bad at geometry , by geometry I mean plane euclidean geometry with similarities and circles. I sometimes feel totally lost with problems. For example:

The parallel sides of trapezoid ABCD are 3 cm and 9 cm(AB and DC).The non parallel sides are 4 cm and 6 cm(AD and BC).A line(PQ) parallel to base(DC) divides the trapezoid into two trapezoids(ABQP and PQCD ) of equal perimeters. Find the ratio in which each of the non parallel sides is divided

I knew that this problem had to do with the Thales theorem but I felt completely lost after equating perimeters.

So my question is can someone recommend a book / videos (anything!) to improve plane geometry? Moreover since I plan on studying Physics, is euclidean geometry needed in Physics?

2. Sep 16, 2015

### micromass

It is occasionally useful, but you don't need much. The basic theorems of triangles are all you'll need. Trig is more important.

But I am absolutely saddened by questions like "Why do I need to know this?" as it implies that if you don't need it, then you'll just ignore it. Euclidean geometry is one of the most beautiful things ever created by humans. It is one of the earliest triumphs of our intellect. Euclid's book influenced all famous scientists up to the 19th century. Newton was so in awe by Euclid's Elements that he did his physics in that style. I don't think it is outrageous to say that the book is in the top 5 most influential science books, together with books like Newton's Principia and Darwin's Origin of Species. But if you truly must find an application for everything you learn, then go ahead and ignore this marvelous piece of work.

3. Sep 16, 2015

### Mastermind01

I want to become better at Euclidean Geometry. It is the only thing in maths (so far! , I'm still in high school) that has completely eluded my grasp. Can you recommend a book / videos (anything!) to improve plane geometry? Should I read Euclid's elements ?

Thanks.

4. Sep 16, 2015

### SteamKing

Staff Emeritus
I wouldn't tackle Euclid until you get better at understanding geometry and geometric concepts.

You say you are still in HS, but have you taken a course in geometry yet?

When you say geometry has "completely eluded my grasp", what does this mean exactly? Are you having trouble understanding the definitions of basic geometric entities, like points, rays, lines, planes, etc.? Or are you having more difficulty constructing or understanding proofs?

In any event, there are a number of texts to choose from and there will be other resources available online.

5. Sep 16, 2015

### micromass

Again, almost every great scientist in history has read Euclid. Some called their experience so profound that their entire perspective on things changed. So is it a good idea to read Euclid? Hell yes.

To speak for myself, I have obtained a degree in math without ever reading Euclid. (so I guess this proves that reading Euclid is not necessary to be successful). But I have recently started it, and the experience is very profound indeed. I really wish I read it when I was still starting my math career. There is a lot of beauty in that book.

But... one must have a decent commentary, because otherwise it might be difficult to appreciate. And some parts might be difficult to understand (don't worry, the majority of the book is very lucid). So if you want to tackle Euclid directly (and I certainly recommend this!), then I suggest the following two books to do it:

1) Euclid's Elements by John Casey. This is really nice since it provides several exercises and things to think about after each Theorem. This book alone is not enough though. And since the book is from 1885, I don't think I'm breaking any copyright by posting links to the free version of the book here: http://bookzz.org/md5/411ebfde439d5f7afb46000dd352b1c4

But this alone is not good enough. You'll need some modern commentary, this is provided by the outstanding book by Hartshorne:

2) Hartshorne "Geometry: Euclid and Beyond". Relevant here is only the first chapter (the rest of the chapters are more difficult, but maybe the first chapter will make you want to read it). It provides very nice commentary and puts Euclid in a modern perspective. I would suggest to use this book as the main text, because it is meant as a companion for Euclid. Contained in the text are directions "Now read theorems 1-20 of Euclid". So read Hartshorne and then read the relevant parts of Euclid when Hartshorne asks you to. Warning though, the exercises in Hartshorne are quite challenging, so don't feel bad if you can't solve many of them.

3) If you don't wish to read Euclid, or if you find the experience to be not so brilliant and profound as I promised, or if you find it too difficult, then I suggest the book by Solomonovich "Euclidean Geometry: A first course". This is meant to be a high school book. It is quite rigorous (especially compared to other high school texts). It follows the spirit of Euclid. It even goes a bit beyond Euclid at places. It has very good problems (In fact, I recommend you getting this book anyway just for the problems!). But I guess it can be a bit childish at times (it's a high school book, so what do you expect, the usual high school book is way more childish however). For example, I'm not sure that an exercise like "find more English words of Greek origin" is really relevant, but this is a really minor point.

Somehow, I feel that a good knowledge of Euclidean geometry will come if you combine all the above three books somehow.

Now, if you are really and truly motivated to study this for the sake of seeing beautiful math and to expand your knowledge, then feel free to message me if you want somebody to mentor you through it. If you just want to get a good grade, or if you just want to know it because it might be useful later on, then don't bother to message me.

6. Sep 16, 2015

### micromass

I do understand your advice, but 100 years ago, Euclid was a standard text in high school classrooms. So if kids 100 years ago could handle it, why can't kids today handle it. Euclid was the first encounter with geometry for a lot of people!

7. Sep 16, 2015

### Mastermind01

What if it's both - expanding knowledge as well getting a good grade ? Will you still help me?

Yup , We're going to finish off with geometry this year - sadly.

I always had trouble understanding planes and yes have trouble especially in constructing proofs.

8. Sep 17, 2015

### micromass

Of course everybody cares about their grades to some extent, so it's not a problem for me.