×

Grade school triangles. (English) Zbl 1210.00044

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.

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
PDFBibTeX XMLCite
Full Text: DOI Link