Circles and polygons
Let us consider the following sequence of circles in a given triangle \(A_1,A_2,A_3\). The circle \(S_1\) is inscribed in the angle \(A_1\); the circle \(S_2\) inscribed in the angle \(A_2\) and tangent to \(S_1;S_3\) inscribed in the angle \(A_3\) and tangent to \(S_2\); etc. The circle \(S_i\) is smaller than the two circles tangent to \(S_{i-1}\). A nice elementary theorem (Money-Coutts theorem) states that the above process is 6-periodic, i.e. \(S_7=S_1\). The author considers the process for parallelograms and shows that the above process is always preperiodic.
scientific article; zbMATH DE number 1485594