Calcut, Jack S. Grade school triangles. (English) Zbl 1210.00044 Am. Math. Mon. 117, No. 8, 673-685 (2010). In grade school, the students use overall triangles witch angle measurements 45-45-90 and 30-60-90. They are right triangles, in which the angles are rational multiples of 180\(^\circ\) and the ratio of the side lengths is at most a square root. The author examines the question: What are these right-angled triangles as well? It is a little surprising that this property only for the 15-75-90 triangle holds. (This triangle is the so-called Achilles’ rectangle.) The proof uses a lot of algebraic tools, in particular number field extensions. The problem could be dealt with qualified students in a seminar group of the secondary school. Reviewer: Erhard Quaisser (Potsdam) Cited in 4 Documents MSC: 00A99 General and miscellaneous specific topics 12F05 Algebraic field extensions 51M04 Elementary problems in Euclidean geometries 97F99 Education of arithmetic and number theory 97G99 Geometry education Keywords:grade school triangles; field extensions PDFBibTeX XMLCite \textit{J. S. Calcut}, Am. Math. Mon. 117, No. 8, 673--685 (2010; Zbl 1210.00044) Full Text: DOI Link