F. van der Blij and the Kletter triangles (Q2831667)

This paper tells us about a very interesting story about a particular type of triangles, from a historical point of view as well as handeling of a problem by several persons, as follows. NEWLINENEWLINEAround 2003, Prof. F. van der Blij (*1923, retired from the University of Utrecht) gave a lecture on so-called Kletter triangles (Th. J. Kletter, a former neighbour of van der Blij, and also a mathematics teacher). The authors of the paper in the Nieuw Archief voor Wiskunde give a very lively story about the ins and outs of the so-called Kletter triangles (an appellation coined by the authors here). NEWLINENEWLINELet us give a Definition: Let \(A\), \(B\), and \(C\) be the corner-points of a triangle \(T\) such that the altitude from \(A\), the bisector of \(B\), and the median of \(C\), intersect all three in one point (i.e. these three lines are concurrent here). Then, such a triangle \(T\) is called a Kletter triangle.NEWLINENEWLINEA Kletter triangle is called rational, if \(|AB|\), \(|BC|\) and \(|CA|\) are rational numbers. Do rational Kletter triangles exist? The paper under review provides the story! The authors mention: \textit{D. L. Mackey}, ``Problem E 374'' [Am. Math. Mon. 46, 168 (1939)] and many other sources, see below, leading to an answer in the affirmative and more. The conchoide of Nicomedes plays a rôle, the cissoide of Diocles also, as well as elliptic curves, in the search for rational Kletter triangles. The authors give a chronological survey of the problem; it turned out recently that the following theorem, due to Charles W. Trigg from 1940, holds:NEWLINENEWLINETheorem (\textit{C. W. Trigg}, ``Solution to Problem E 374'' [Am. Math. Mon. 47, 176 (1940)]). Three positive numbers \(a\), \(b\), \(c\) with \(a+b=c\) do occur as the three lengths of the sides in a Kletter triangle precisely then if \(b^2=a^2-4a-4\) holds. After an eventual rescaling with a positive factor, all Kletter triangles do appear. (For more information on rational Kletter triangles, see [\textit{R. K. Guy}, Am. Math. Mon. 102, No. 9, 771--781 (1995; Zbl 0847.11031)].)NEWLINENEWLINEThere is more to tell about the authors' paper.NEWLINEHere we confine ourselves in mentioning relevant literature as given in the paper [\textit{E. Bakker} et al., Expo. Math. 34, No. 1, 82--94 (2016; Zbl 1397.11088); \textit{J. Top}, [``Albime triangles'', Colloquium lecture, Utrecht (2014), \url{http://www.math.rug.nl/~top/lectures/Utrechtalbime.pdf}].NEWLINENEWLINELet me finish this review with the remark, that non-Dutch speaking mathematicians should consult this source under review, and try to make clear for themselves, the (hi)story about Kletter triangles; it is worth to do so!