On the Fermat geometric problem (Q2871648)

Fermat's geometric problem, or theorem, alluded to in the title of the paper under review refers to a theorem that Fermat announced, without proof, in order to demonstrate to the English mathematicians of his time that he was well-versed not only in number theory but in geometry as well. The underlying configuration of the theorem consists of a rectangle \(ABB'A'\), a semicircle having \(AB\) as a diameter and drawn outwardly, and an arbitrary point \(P\) on the semicircle. Denoting the points where \(PA'\) and \(PB'\) cross \(AB\) by \(C\) and \(D\), respectively, and letting \(m = AB/AA'\), Fermat's theorem states that if \(m=\sqrt{2}\), then \(| AD|^2+| BC|^2 =| AB|^2\).NEWLINENEWLINENEWLINEThe author of the paper under review starts with a history and an interesting list of references containing the several existing proofs of Fermat's theorem, including a proof by Euler. After giving one of these proofs, together with a proof of the converse, he moves on to investigate properties of the configuration when the ratio \(m\) is arbitrary, proving as many as 18 theorems, and discovering interesting values of \(m\) other than \(\sqrt{2}\). A confusing feature of the paper are the definitions of \(A''\), \(B''\), \(P'\), \(C'\), and \(D'\) at the end of Section 1 (p. 138). While the definitions of \(A''\), \(B''\), \(P'\) give the impression that \(Q'\) is to denote the reflection of \(Q\) in the line \(AB\) for any point \(Q\), the definitions of \(C'\) and \(D'\) come to contradict this. The readability of the paper would be enhanced and made more comfortable if this possible ambiguity was avoided and if a figure illustrating where these points lie was added. NEWLINENEWLINENEWLINEThis reviewer would like to add to the list of references the fascinating book [Geometry by its history. Berlin: Springer (2012; Zbl 1288.51001)] by \textit{A. Ostermann} and \textit{G. Wanner}, where Section 6.6 (pp. 170--171) is devoted to Fermat's theorem. He would also like to raise the question regarding the reasons that led Fermat to consider this configuration and to discover this theorem.