On nested sequences of triangles (Q1035231)

For a triangle \(T\) in the Euclidean plane and a real number \(s\) let \(M_s(T)\) denote the triangle whose vertices divide the sides of \(T\) in the ratio \(s:1-s\); \(M_s(T)\) is called the \(s\)-medial triangle of \(T\), and triangles \(M_s^n(T)\), \(n=1,2,\dots\), are defined inductively. The author investigates the values of \(s\) for which the sequence \(M_s^n(T)\) converges in shape. He shows that if \(T\) is not equilateral, then \(s=1/2\) is the only such value, and he characterizes also those values of \(s\) for which the sequence of shapes is periodic, and for which it is everywhere dense. Besides this, a result of Emmerich is elegantly reproved. It says that the Brocard angle of \(M_s^n(T)\) is the same as that of \(T\), and that each triangle having the same Brocard angle as \(T\) is similar to \(M_s(T)\) for a unique value \(s\). Similar investigations are done for analogously derived \(s\)-Routh triangles of \(T\) which are enclosed in cevians generated by the same side-dividing points. Shape functions occur as the basic tool used by the author.