Reflection triangles and their iterates (Q2880800)

The reflection triangle \(\sigma (T)\) of a triangle \(T\) is the triangle whose vertices are the reflections of the vertices of \(T\) about the opposite sides. This definition was introduced by \textit{J. van Ijzeren} in [EUT-Rep., Eindhoven 84-WSK-03, 356--373 (1984; Zbl 0548.51015)], where the seemingly easy question whether \(T\) can be recovered from \(\sigma (T)\) is investigated and seen to lead to an equation of degree 7. Chapter 16 of \textit{O. Bottema}'s book [Topics in elementary geometry. New York, NY: Springer (2008; Zbl 1159.51006)] is devoted to the, again seemingly easy, problem of characterizing those \(T\) for which \(\sigma (T)\) is degenerate. Both results are non-trivial.NEWLINENEWLINEIn view of the above, one would expect the behavior of the sequence \(\sigma^n (T)\) of triangles (or rather their shapes) to be quite complicated to determine. This is indeed so, and this is what is done in the paper under review, where the author determines those \(T\) for which this sequence is convergent, periodic, chaotic, etc., and where many questions regarding this sequence are answered. As expected, the paper is long and technical. It contains as many as 20 non-trivial theorems spread over 47 pages and answering almost any question that comes to mind regarding the afore-mentioned sequence \(\sigma^n (T)\). For example, a sample theorem, Theorem 13, shows that if \(T\) is acute or right, then \(\sigma^n (T)\) converges in shape to an equilateral triangle.