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I have unsuccessfully attempted several times over the years to formalize high-school (Euclidean) geometry, or even a working subset of it. Think very simple, diagramless geometry.

The usual two-column proofs are quite inadequate for this. Any suggestions would be appreciated. My intended audience is advanced high-school students and first or second year undergrads. As such, the axioms of Tarski or Hilbert would not be helpful. Maybe a workable subset of them to address a limited set of problems.

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  • $\begingroup$ I have not read it, but this looks promising: bookstore.ams.org/view?ProductCode=AMSTEXT/51 $\endgroup$ Commented Feb 18, 2023 at 16:20
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    $\begingroup$ Can you clarify what is meant by "diagramless" here? It's worth investigating any other geometry books in the Sally series, I know of at least one other. The series books are typically excellent. $\endgroup$
    – Opal E
    Commented Feb 18, 2023 at 16:26

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Clark and Pathania might be of interest to you.

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"This textbook provides a full and complete axiomatic development of exactly that part of plane Euclidean geometry that forms the standard content of high school geometry. It begins with a set of points, a measure of distance between pairs of points and ten simple axioms. From there the notions of length, area and angle measure, along with congruence and similarity, are carefully defined and their properties proven as theorems. It concludes with a proof of the consistency of the axioms used and a full description of their models. It is provided in guided inquiry (inquiry-based) format with the intention that students will be active learners, proving the theorems and presenting their proofs to their class with the instructor as a mentor and a guide.

The book is written for graduate and advanced undergraduate students interested in teaching secondary school mathematics, for pure math majors interested in learning about the foundations of geometry, for faculty preparing future secondary school teachers and as a reference for any professional mathematician. It is written with the hope of anchoring K-12 geometry in solid modern mathematics, thereby fortifying the teaching of secondary and tertiary geometry with a deep understanding of the subject."

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