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There are books about counterexamples in analysis, topology or probability. Is there any book that focuses on counterexamples in plane geometry or loopholes in geometrical proofs?

I am particularly interested in books that illustrate "honest" false proofs of high-school geometry statements, something like Lewis Carroll's proof that every obtuse angle is equal to a right angle. I vaguely remember that there was a geometry book that discussed, in a small section, how some geometrical proofs contain loopholes because the proof writers forget about the concept of betweenness, but I can't recall the title.

Edit. After browsing on amazon.com for hours, I found a book that might be relevant:

A.I. Fetisov and Ya. S. Dubnov, Proof in Geometry: With "Mistakes in Geometric Proofs" (Dover/amazon.com)

According to an MAA review, this is actually a compilation of two separate books, Proof in Geometry by Fetisov and Mistakes in Geometric Proofs by Dubnov. The review is only neutral, and I have yet to see if the proofs are good (or should I say "bad"?) enough.

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Theorems and Counterexamples in Mathematics by Gelbaum and Olmsted has a section on plane Euclidean geometry

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  • $\begingroup$ Thanks for the information. As the section contains only two examples, I'll see if I can borrow a copy of this book. $\endgroup$ – user1551 Feb 18 at 15:59

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