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I have been looking at questions on Math Stack Exchange and I am frequently coming across topics that sound as if they could have been optional chapters in a high school geometry class, but I have never heard of them. "Radical circles", "Euler line", and so forth. It feels almost as if there were a Geometry II class that I somehow skipped, leaving a hole in my knowledge.

Note I am not looking for advanced courses like differential geometry or projective geometry, but rather things that look like challenge problems in ordinary high school geometry. In particular I would like to find a textbook for self-study. I own several high school "Geometry I" textbooks already. I would like a textbook for this hypothetical "Geometry II" course. Any suggestions are appreciated.

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    $\begingroup$ People rave about Elementary Geometry from an Advanced Standpoint, 3rd Edition, but I don't own a copy (it is quite expensive). Maybe someone who has used/read it can remark. $\endgroup$
    – Joseph O'Rourke
    Apr 4, 2021 at 16:11
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    $\begingroup$ @Joseph_O'Rourke, I think the standpoint is very advanced. My guess is, that book isn't quite right for this request, which is asking for topics. $\endgroup$
    – Sue VanHattum ♦
    Apr 5, 2021 at 1:55
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    $\begingroup$ |n some very old geometry texts you may see AB:CD::EF:GH which means AB/CD=EF/GH. $\endgroup$
    – DanielWainfleet
    Apr 9, 2021 at 2:02

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I recommend either Excursions in Geometry by Ogilvy or Geometry Revisited by Coxeter and Greitzer. Both are cheap too.

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    $\begingroup$ +1. Also Coxeter's larger book Introduction To Geometry. $\endgroup$
    – DanielWainfleet
    Apr 5, 2021 at 18:07
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    $\begingroup$ This great old book has a wealth of material. I was hoping it wpuld be cheaper. amazon.ca/Treatise-Plane-Advanced-Trigonometry/dp/0486441776 $\endgroup$
    – DanielWainfleet
    Apr 5, 2021 at 22:12
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    $\begingroup$ @DanielWainfleet: I got my copy, an unused 1957 Dover reprint (probably printed in the mid to late 1960s) for $3.00 at this bookstore sometime in the late 1970s. I know it was \$3.00 because there are two tiny white \$3.00 stickers still on my copy saying "PUBLISHERS LIST PRICE", and that bookstore always (at least back then) only charged for Dover books what the cover price showed, even if very low due to the book being an old Dover printing. $\endgroup$
    – Dave L Renfro
    Apr 7, 2021 at 18:36
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    $\begingroup$ By the way, I assume you know digital versions of Hobson's book are freely available on the internet, some (all?) being digital copies of editions published prior to when the Dover reprint appeared. $\endgroup$
    – Dave L Renfro
    Apr 7, 2021 at 18:40
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    $\begingroup$ @DanielWainfleet: Pretty much anything published in the 1800s can be found online (e.g. see this), and the first two editions of Hobson appeared before 1900. However, Hobson's book is definitely sufficiently well known for some later editions to have been picked up by archive.org and probably other places as well. Those reading who are interested in advanced trigonometry texts, see the comments here. $\endgroup$
    – Dave L Renfro
    Apr 8, 2021 at 9:09
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In my (lack of) experience, the second class in Euclidean geometry is actually an undergraduate course that seems to be often called College Geometry. And, yeah, there are so many fascinating topics there that are accessible to gifted HS students that don't really measure up to other topics when it comes to career and college readiness, but they were the pinnacle of mathematical and logical knowledge before calculus was developed.

Something that is more readable than a textbook but just as informative that I really loved was The Secrets of Triangles by Posamentier. There aren't formal exercises, but there are plenty of claims in the narrative that you can verify for yourself on a sheet of scrap paper. I also read a college geometry textbook by Posamentier that I really appreciated if you are particular about a less casual narrative style.

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A quick Google search returned the following texts based on key words from your posting: rational circle, Euler line, geometry textbook. Based on skimming the contents and prefaces, I think they are good chance to match your desires:

  1. A Beautiful Journey through Olympiad Geometry, Stefan Lozanovski, 2016

https://www.olympiadgeometry.com/the-book.html

  1. Euclidean Geometry in Math Olympiads, Evan Chen, 2016

https://web.evanchen.cc/geombook.html

At least the latter link seems to have additional resources to the named book.

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I took an undergraduate course in "advanced planar geometry" in preparation for secondary teaching. It was from the text by Isaacs, "Geometry for College Students (Pure and Applied Undergraduate Texts)" and is available for purchase here. There are also less upstanding ways to obtain this text.

It appears to cover all the topics you've mentioned: the Euler line is the topic of 2.C and radical circles in 3.D.

Other topics of note that I enjoyed: the nine-point circle, discussion of Simson lines, the butterfly theorem, and Ceva's theorem, and finally the discussion of "constructible numbers" at the end.

The text should be accessible to anyone with a basic mathematical maturity with proof, ideally at least introductory proof at the college level, but it re-introduces "Geometry I" in its first chapter, slightly more formally.

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I wonder if you'd enjoy an older textbook on Solid Geometry. (The only ones I've ever seen are quite old. Maybe there's newer.) But that may not address the particular topics you mentioned.

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  • $\begingroup$ I am definitely a fan of using older textbooks. They sometimes have a lot of lost lore. $\endgroup$
    – RobertTheTutor
    Apr 5, 2021 at 2:04
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So some great books on geometry and written for middle/high schoolers are those written by Kiselev and translated to English by Alexander Givental. The textbooks are "Planimetry" and "Stereometry" respectively. Maybe try taking a look in either of those books and see if there is something that suits you?

Here's a link to the first.

https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202

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@MatthewDaly's mention of The Secrets of Triangles reminded me of the just released A Cornucopia of Quadrilaterals, which I've been reading. (Alsina, Claudi, and Roger B. Nelsen. Vol. 55. American Mathematical Soc., 2020.)

For example, for a bicentric quadrilateral (cyclic and tangential) of side lengths $a,b,c,d$ and angles $A,B,C,D$: \begin{align*} a + c &= b + d \\ A + C &= B + D \end{align*}

CoverImage

And now a sequel of sorts, A Panoply of Polygons (AMS link):

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A very unique kind of book is “Machine Proofs in Geometry: Automated production of readable proofs for geometry theorems” by Shang-Ching Chou, Xiao-Shan Gao and Jing-Zhong Zhang. (It is of interest to people studying automated reasoning as well).

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