What geometry book would be a good follow-up to AoPS'S Intro to Geometry?

[MBM]

My son wants to study more advanced geometry but is not sure which book would be good. He finished all of AoPS geometry awhile ago and is working through Challenging Problems in Geometry by Alfred Posamentier and Charles Salkind, but he was wondering if someone might know of another good text to follow AoPS's.

 

Challeging Problems in Geometry covers

 

SECTION 1

 

Congruence and Parallelism

Triangles in Proportion

The Pythagorean Theorem

Circles Revisited

Area Relationships

 

SECTION 2

 

A Geometric Potpourri

Ptolemy and the Cyclic Quadrilateral

menelaus and Ceva: Collinearity and Concurrency

The Simson Line

The Theorem of Stewart

 

His goal is to make USAMO in the next four years if that helps, so I guess he would like something geared toward contest math.

 

Any help is much appreciated! :)

[Kathy in Richmond]

My son wants to study more advanced geometry but is not sure which book would be good. He finished all of AoPS geometry awhile ago and is working through Challenging Problems in Geometry by Alfred Posamentier and Charles Salkind, but he was wondering if someone might know of another good text to follow AoPS's.

 

Challeging Problems in Geometry covers

 

SECTION 1

 

Congruence and Parallelism

Triangles in Proportion

The Pythagorean Theorem

Circles Revisited

Area Relationships

 

SECTION 2

 

A Geometric Potpourri

Ptolemy and the Cyclic Quadrilateral

menelaus and Ceva: Collinearity and Concurrency

The Simson Line

The Theorem of Stewart

 

His goal is to make USAMO in the next four years if that helps, so I guess he would like something geared toward contest math.

 

Any help is much appreciated! :)

 

Well, there's a need for AoPS to hurry up and write an intermediate geometry text of their own soon! :)

 

My kids used a combination of the book you mentioned and also Geometry Revisited by Coxeter and Greitzer & Plane Euclidean Geometry by Gardiner and Bradley to fill in the intervening years between introductory geometry and AoPS Olympiad Geometry online course. They both took that online course from Richard R in grade 11/12 & would highly recommend it.

 

Honestly, though, what most improved their geometry abilities in the intermediate years was just doing harder and harder geometry problems. They used sources such as past AIME exams, USAMTS contests (probably one of the best sources for my two), and old HMMT geometry rounds. That and being on a math team for dd and going to summer math camps for both of them.

Edited November 6, 2011 by Kathy in Richmond

[MBM]

Thank you, Kathy! Your suggestions are always so helpful, and I know these will give him something to work on.

 

:)

[jeri]

DD (14) is using it for her Geom course. (Tried LoF and did *not* like it!). So, is this enough for a std high school Geom class? (She liked algebra much better and really isn't too fond of Geom.)

 

Thanks.

 

jeri

[Kathy in Richmond]

DD (14) is using it for her Geom course. (Tried LoF and did *not* like it!). So, is this enough for a std high school Geom class? (She liked algebra much better and really isn't too fond of Geom.)

 

Thanks.

 

jeri

 

Don't worry at all, jeri! AoPS geometry is more than enough. In fact, it's a lot more than enough. :001_smile:

 

What we're talking about here is what to do next if a student is interested in pursuing geometry study at higher levels, whether for Olympiad prep or just for pure fun & knowledge.

[jeri]

jeri

[quark]

Reviving an old thread to ask about Coxeter's Geometry Revisited.

 

Kathy or anyone else, do you feel that Coxeter could be a good follow up to any high school geometry course, and not just AoPS?

 

DS will be finishing Jurgensen's Geometry in a few months and his tutor has suggested starting him on Coxeter's book next.

 

I can't figure out where this book fits into the usual math sequence (or maybe it doesn't?). Does Coxeter's book overlap in any way with other high school math for e.g. Algebra 2?

 

And generally, how difficult is Coxeter's book? The ToC makes my head swim.

 

Thank you.

[kiana]

It does not fit into the standard sequence. With the possible exception of some material on conic sections that may be covered in precalculus, there is virtually no overlap.

 

It is, however, an excellent choice for a young, gifted child (ten? impressive!) who is accelerated in mathematics and may be looking at mathematics competitions or studying it seriously later on. I have not used it myself.

 

Suggestion: Get the book now, since the tutor's recommended it and wants to try it. If he's not ready for it yet, put it back on the shelf and do something else until he is. (AOPS's discrete math courses should definitely be at a good level for him as well.)

[quark]

Thank you kiana! You have been so helpful to me numerous times in the past. :001_smile:

[Sebastian (a lady)]

Well, there's a need for AoPS to hurry up and write an intermediate geometry text of their own soon! :)

 

My kids used a combination of the book you mentioned and also Geometry Revisited by Coxeter and Greitzer & Plane Euclidean Geometry by Gardiner and Bradley to fill in the intervening years between introductory geometry and AoPS Olympiad Geometry online course. They both took that online course from Richard R in grade 11/12 & would highly recommend it.

 

Honestly, though, what most improved their geometry abilities in the intermediate years was just doing harder and harder geometry problems. They used sources such as past AIME exams, USAMTS contests (probably one of the best sources for my two), and old HMMT geometry rounds. That and being on a math team for dd and going to summer math camps for both of them.

 

I found a copy of this book (Spherical Trigonometry with Naval and Military Applications) at a used book store. It has delightful problems involving navigation, paratroopers, naval gunfire, aircraft and tanks. The authors were math instructors at the US Naval Academy before and during the war. This is a special edition of the math book they'd already written with the military oriented problems added, I think probably for use in OCS or other training programs. There also seems to be a Geometry with Military and Naval Applications title.

 

ETA: (Wow, sometimes I love the internet) I found an online copy of the Geometry book. One thing I liked about the trig book I have (and what I like about AoPS and Dolciani) is that they manage to move the math out of the abstract and show situations where what you learned is applicable to the world.

 

(BTW, I find it ironic that I never even heard of this book while at the Naval Academy. I guess we were considered beyond these types of problems. The calculus class with Major C~ that I stumbled through could sure have used some practical insights. I wish they'd revive this title.)

Edited October 23, 2012 by Sebastian (a lady)

[Kathy in Richmond]

Reviving an old thread to ask about Coxeter's Geometry Revisited.

 

Kathy or anyone else, do you feel that Coxeter could be a good follow up to any high school geometry course, and not just AoPS?

 

DS will be finishing Jurgensen's Geometry in a few months and his tutor has suggested starting him on Coxeter's book next.

 

I can't figure out where this book fits into the usual math sequence (or maybe it doesn't?). Does Coxeter's book overlap in any way with other high school math for e.g. Algebra 2?

 

And generally, how difficult is Coxeter's book? The ToC makes my head swim.

 

Thank you.

 

Hey, quark! How cool that your son's tutor wants to work through Coxeter with him. It's a very challenging book & a classic in the field, but could work with the guidance he's offering you. :)

 

Based on my experience, though, I have some other ideas (my kids used most of these resources before digging into Coxeter later in high school):

 

I would revisit the geometry chapters in the AoPS classic volumes (even volume 1 has some challenging geom topics beyond most standard curricula) or the AoPS geometry text. He could work through those as an intermediate step before Coxeter (especially hitting the challenging problems since he's had Jurgensen already)

 

I also very much like the Gardiner & Bradley Euclidean geometry book that I mentioned in a previous post. It's written to UK kids who are preparing for math olympiads there & has a more student feel to it. I have both books in front of me right now, and Coxeter is definitely denser and a more challenging read. Gardiner will take him through many of the same topics as Coxeter. I think that the problem sets in Gardiner are more accessible than those in Coxeter.

 

Another totally different approach would be to work through Robin Hartshorne's Geometry: Euclid and Beyond. Mathwonk has his very helpful Epsilon camp notes on his website to go along with this approach (see the link in his WTM signature). With a personal mentor, your son could have the experience that we offered the math kids at Epsilon camp.:)

 

My last thought (what I'd do personally with a kid like you have :D ) would be to work through one or more of Gelfand's texts next. His trigonometry is VERY geometrically motivated. My kids loved it at that stage of the game, and it's very accessible to advanced young kids with a mentor to help.

 

Just some rambling thoughts...your son may very well not need these & may be happy with Coxeter!

[Kathy in Richmond]

I found a copy of this book (Spherical Trigonometry with Naval and Military Applications) at a used book store. It has delightful problems involving navigation, paratroopers, naval gunfire, aircraft and tanks. The authors were math instructors at the US Naval Academy before and during the war. This is a special edition of the math book they'd already written with the military oriented problems added, I think probably for use in OCS or other training programs. There also seems to be a Geometry with Military and Naval Applications title.

 

ETA: (Wow, sometimes I love the internet) I found an online copy of the Geometry book. One thing I liked about the trig book I have (and what I like about AoPS and Dolciani) is that they manage to move the math out of the abstract and show situations where what you learned is applicable to the world.

 

(BTW, I find it ironic that I never even heard of this book while at the Naval Academy. I guess we were considered beyond these types of problems. The calculus class with Major C~ that I stumbled through could sure have used some practical insights. I wish they'd revive this title.)

 

Cool! I have an old copy of a book my dad studied in high school (WW II years) called Aviation Mathematics. The math & science subjects he studied in the 1940s in a rural PA high school (i have his chem and physics texts also) were SO much more applied than today's courses.

[Sebastian (a lady)]

I think it's ironic (and sad) that modern math books have gone to a "tell, don't show" method of making math "relevant". Wha I mean is that they have full color photos and bios of people who use math, but they often lack the interesting applied problems , ie word problems that let the kid try their own hand at putting the math to use.

 

One reason I love texts with word problems is the feeling of power that comes with solving them. And I like books that preserve math terms and symbols because of the sense of initiation into a crypto society.

[quark]

I found a copy of this book (Spherical Trigonometry with Naval and Military Applications) at a used book store. It has delightful problems involving navigation, paratroopers, naval gunfire, aircraft and tanks. The authors were math instructors at the US Naval Academy before and during the war. This is a special edition of the math book they'd already written with the military oriented problems added, I think probably for use in OCS or other training programs. There also seems to be a Geometry with Military and Naval Applications title.

 

ETA: (Wow, sometimes I love the internet) I found an online copy of the Geometry book. One thing I liked about the trig book I have (and what I like about AoPS and Dolciani) is that they manage to move the math out of the abstract and show situations where what you learned is applicable to the world.

 

(BTW, I find it ironic that I never even heard of this book while at the Naval Academy. I guess we were considered beyond these types of problems. The calculus class with Major C~ that I stumbled through could sure have used some practical insights. I wish they'd revive this title.)

 

I love free! There's a free online copy of Coxeter's Geometry Revisited too! :001_smile:

[quark]

My last thought (what I'd do personally with a kid like you have :D ) would be to work through one or more of Gelfand's texts next. His trigonometry is VERY geometrically motivated. My kids loved it at that stage of the game, and it's very accessible to advanced young kids with a mentor to help.

 

Just some rambling thoughts...your son may very well not need these & may be happy with Coxeter!

 

Thank you Kathy!

 

The Gardiner and Bradley book is pricey! I'll check my library for sure. And we have AoPS Vol 1 still unused. Time to bring it out!

 

Gelfand has been on my mind a lot lately too so thank you for suggesting that. Just previewed the Trig on Amazon more carefully...you are right, Kathy. I think he will love it! Perhaps more than AoPS Vol 1.

 

His tutor has an uncanny ability to read DS and thinks Coxeter will be the perfect choice and I've learned to trust his suggestions. Since we are in no hurry, I don't mind if DS spends years on it. Still thinking carefully though as I love to research options and am grabbing this opportunity to think aloud with you math moms! :001_smile:

 

DS likes AoPS Algebra very much (he tried the Geometry book a year+ ago but preferred Jurgensen). I think he will like Gelfand very much too. If DS worked on Gelfand's Algebra and Trig books with some additional problems from AoPS Algebra...all alongside a stretched-out Coxeter, do you think he'll need another formal course in Algebra 2? He will effectively be doing Algebra 2 and Coxeter's Geometry concurrently right?

 

In addition, I have no idea how to document this. Jurgensen's is obviously a high school geometry text and DS works hard on the C problems, some of which are enough to turn my puny brain inside out. Based on DS's work, I think his tutor might call it an Honors Geometry course. If he goes ahead with Coxeter what should I call it?

[Kathy in Richmond]

The Gardiner and Bradley book is pricey! I'll check my library for sure. And we have AoPS Vol 1 still unused. Time to bring it out!

 

Gelfand has been on my mind a lot lately too so thank you for suggesting that. Just previewed the Trig on Amazon more carefully...you are right, Kathy. I think he will love it! Perhaps more than AoPS Vol 1.

 

His tutor has an uncanny ability to read DS and thinks Coxeter will be the perfect choice and I've learned to trust his suggestions. Since we are in no hurry, I don't mind if DS spends years on it. Still thinking carefully though as I love to research options and am grabbing this opportunity to think aloud with you math moms! :001_smile:

 

Sounds like a good plan! Coxeter would provide lots of learning if taken at the right pace. I just like to throw out other ideas; it's always fun and useful to have an extra book on the shelf for reference.:) I bought my copy of Gardiner through the AoPS bookstore. When I just checked, I found that they're out of stock currently. I believe it was about $25 or so.

 

DS likes AoPS Algebra very much (he tried the Geometry book a year+ ago but preferred Jurgensen). I think he will like Gelfand very much too. If DS worked on Gelfand's Algebra and Trig books with some additional problems from AoPS Algebra...all alongside a stretched-out Coxeter, do you think he'll need another formal course in Algebra 2? He will effectively be doing Algebra 2 and Coxeter's Geometry concurrently right?

 

No, I wouldn't bother with a routine Alg 2 course in his case. I did something similar with my dd. On her transcript i listed an "algebra 2" year, & in the course description section, I simply listed the resources we used that year.

 

In addition, I have no idea how to document this. Jurgensen's is obviously a high school geometry text and DS works hard on the C problems, some of which are enough to turn my puny brain inside out. Based on DS's work, I think his tutor might call it an Honors Geometry course. If he goes ahead with Coxeter what should I call it?

 

I think it would be fine to call your son's Jurgensen year Honors Geometry. Then, for his Coxeter work, I'd probably call it Advanced Geometry and write a course description paragraph. You might want to see the AoPS Olympiad Geometry course description for ideas.

 

Then have his tutor write one of his letters of recommendation.:)

[quark]

Kathy, you are incredibly helpful always. Thanks so much!

[quark]

I think it's ironic (and sad) that modern math books have gone to a "tell, don't show" method of making math "relevant". Wha I mean is that they have full color photos and bios of people who use math, but they often lack the interesting applied problems , ie word problems that let the kid try their own hand at putting the math to use.

 

One reason I love texts with word problems is the feeling of power that comes with solving them. And I like books that preserve math terms and symbols because of the sense of initiation into a crypto society.

 

I've been thinking about this. What is helping here is to pair the math with the appropriate level in science. DS enjoys applying his algebra in his physics course and figuring out interesting tidbits such as why it's safer to park a car on a slope of angle x vs at a steeper slope. He doesn't have the practical experience yet being too young to drive (although he could experience it as a passenger) but I'm glad he has a chance to see the math behind the physics and make some sense of it. And as he moves into higher levels in the future, I'm glad he has the math to help it all come together. Just learning the math alone, while great, wouldn't have been as effective for real-life understanding/ application.

[Tattarrattat]

Thanks Kathy, quark and everyone else for sharing the resources. DS also needs more geometry books after taking AoPS geometry class.

 

Gardner and Bradley's Plane Euclidean Geometry is now back in stock on AoPS website, in case anyone wants to order it. Much cheaper than Amazon.

 

http://www.artofproblemsolving.com/Store/viewitem.php?item=ukmt:plane

[quark]

:hurray: Thank you Tattarrattat! Just bought one.

[LisaKinVA]

I found a copy of this book (Spherical Trigonometry with Naval and Military Applications) at a used book store. It has delightful problems involving navigation, paratroopers, naval gunfire, aircraft and tanks. The authors were math instructors at the US Naval Academy before and during the war. This is a special edition of the math book they'd already written with the military oriented problems added, I think probably for use in OCS or other training programs. There also seems to be a Geometry with Military and Naval Applications title.

 

ETA: (Wow, sometimes I love the internet) I found an online copy of the Geometry book. One thing I liked about the trig book I have (and what I like about AoPS and Dolciani) is that they manage to move the math out of the abstract and show situations where what you learned is applicable to the world.

 

(BTW, I find it ironic that I never even heard of this book while at the Naval Academy. I guess we were considered beyond these types of problems. The calculus class with Major C~ that I stumbled through could sure have used some practical insights. I wish they'd revive this title.)

 

My husband HAS this book from his days at USMMA! I've got it sitting on the sidelines for my son to look through after he finishes AoPS.

[MBM]

AoPS also offers Olympiad Geometry classes online. My son took the class last summer but wasn't able to work through them all at the time, so he downloaded the problem sets, class discussions, and answers to work on later. Just throwing that idea out there for those who might want another resource.

[Sebastian (a lady)]

I'm impressed, though I probably shouldn't be. The USMMA guys I knew could also do celestial sight reductions without the strip telling them what step came next.