# Introduction to Geometry MOBI ¼ Introduction to

Textbookpages,Solutions Manualpages, For context, I m pretty involved in math competitions and have a safe index for the USAJMO this year So this is coming from a competition math perspective If your question is I m a school teacher, should I be using this book in my Honors Geometry class then the answer is definitely yes This book is probably much better than any alternative you re considering But if your question is I want to personally get good at geometry for math competitions, that s when I can t exactly recommend thi For context, I m pretty involved in math competitions and have a safe index for the USAJMO this year So this is coming from a competition math perspective If your question is I m a school teacher, should I be using this book in my Honors Geometry class then the answer is definitely yes This book is probably much better than any alternative you re considering But if your question is I want to personally get good at geometry for math competitions, that s when I can t exactly recommend this book.I didn t rate this book poorly because it s a bad book, by any means But this is probably one of the most overrated math textbooks out there It s not bad, it s in fact far above average and I would definitely prefer this over any school textbook It has competition problems too, which aligns with my own philosophy of you can only remember hard stuff by doing hard problems It s also a decent place to pick up the basics, but the book s structure doesn t lend itself to it In fact, to use this book optimally, you probably have to actively go against the design of this book.The pacing of the book is very slow and it s very hard to slog through Unless you re a complete beginner to competition geometry, you re probably better off just picking stuff up from doing problems That s not to say you can t pick up problems from doing them in the book the book has a quite decent selection of them But actually reading the text should be treated as a last resort This is what my opinion is in general, but in this case I feel it muchstrongly This might be due to personal preference, but the structuring of the book is terrible Things that do not deserve to be chapters are artificially inflated in length, and chapter labeling tries to be clever at the expense of clarity Specific complaint Special Parts of Triangles otherwise known as Triangle Centers or Cevians, both of those would ve been fine And the book takes up far too much time and space with stuff that really needs to be stated succinctly A perpendicular bisector of a segment is the locus of points equidistant from its endpoints, and An angle bisector of two lines is the locus of points equidistant from those two lines The proofs are also one liners Notice triangle X is congruent to triangle Y And this kind of stuff needs to be made succinct to stand out The use of one way proofs instead of just a biconditional argument that encompasses the whole thing is also terribly inefficient and teaches bad habits There are some USAMO IMO problems where proving one direction and proving the other are actually two separate non trivial tasks, but this is rare and being able to biconditionally prove something with one sweeping argument is still good And worst of all, there s no way to tell what the most important facts are The four triangle centers and the four cevians Perpendicular bisector isn t a cevian, but my point stands In a good manuscript I should be able to, without effort, tell what the triangle centers are, and the gist of their existence property proofs For this one it s very hard to tell.The book also treats people like idiots, maybe because it s just geometry standard even though nobody actually good at math cares what the standard is The book insists on saying things like SAS congruence or HL congruence or whatever Stuff that can be and should be one liners aren t What this book doesn t seem to understand is this You don t need to spell out all of the trivial details Just leave enough for the reader to be able to somewhat quickly pick it up on their own Well known theorems are presented as problems, so the fact that they re well known and the fact that they aren t novel interesting, just necessary to build upon, is not communicated at all to the reader.Even though I harp on this book, I realize that the reason it s so praised is in part because it deserves to be It s also very hard to write a good introduction to geometry my attempts have also ended up being muchsuited to people who already have some experience though I do try to make my manuscript beginner friendly So I think the world is made a much better place with this book rather than without, because this book fills a void that really needs to be filled But it s not the holy grail and it s not the only way to get good at Geometry.What do I recommend instead I recommend doing lots of hard 2D geometry problems You ll notice a pattern at least in the AMC AIME, it s not proving some triangle is similar to some other triangle or some angle is congruent to some other angle In fact that part is usually trivial The hard part is actually noticing what you need to prove, and convincing yourself that it s true instead of convincing yourself of some other stupid things Consider PUMAC 2016 G7 it s a very hard problem which is why it s the second to last problem, after all , but not because we use some very obscure theorem that you have to read 100 textbooks to memorize It s because the similar triangle sorry, spoilers is very well hidden What s evenamazing is that problems like these are one liners The gist of the solution to really, really hard geometry problems can just be summarized in 4,5 or 6 lines, which is really impressive considering how long harder Algebra Geometry Combinatorics problems are.You might also hear people say that geometry is the most theory heavy subject This is true, in the context of the IMO There s also lots of theory in computational geometry, but AoPS Geometry doesn t even get a pass here the stuff it presents is artificially inflated to look likethan it is, and the real meat doesn t even get included in there despite how easily and naturally it could be Take Radical Axes for example at least tell everyone that the common chord of two circles bisects their common external tangent In summary This book is a good starting point for beginners, but it should not be treated as the holy grail and you should try to get away with using it as little as possible This is the best high school geometry curriculum I ve ever come across, and one of the only ones designed specifically for gifted math students Topics are introduced using a discovery approach a problem is posed for the student to attempt on his or her own Next, a solution is provided for clarity In the process, theorems are discovered, which are then immediately used in solving the next problem s , and so on This is exactly the reverse approach that most geometry textbooks take introduci This is the best high school geometry curriculum I ve ever come across, and one of the only ones designed specifically for gifted math students Topics are introduced using a discovery approach a problem is posed for the student to attempt on his or her own Next, a solution is provided for clarity In the process, theorems are discovered, which are then immediately used in solving the next problem s , and so on This is exactly the reverse approach that most geometry textbooks take introducing theorems first, and only then showing why they are true which really takes most of the fun and creativity of geometry out of the learning process The discovery approach wouldn t work unless it were masterfully crafted, and indeed, the sequence of problems is no less than brilliant Use this with your brightest math students and watch their faces light up Geometry is often taught in a memorize the theorems approach and leaves it up to the students to attempt to solve problems usually on the easier spectrum , absolutely not the case here As expected from AoPS, the students are expected to attempt to discover the theorems and problem solving processes themselves using carefully crafted problems which guide the students into doing so This leads a muchdeep understanding of geometric properties and theorems than what would come out of the a Geometry is often taught in a memorize the theorems approach and leaves it up to the students to attempt to solve problems usually on the easier spectrum , absolutely not the case here As expected from AoPS, the students are expected to attempt to discover the theorems and problem solving processes themselves using carefully crafted problems which guide the students into doing so This leads a muchdeep understanding of geometric properties and theorems than what would come out of the average geometry curriculum Simply put the most amazing book on geometry I ve ever come across.P.S AoPS discourages those stupid two column proofs which is an absolute win