A Sangaku-type problem with regular polygons, triangles, and congruent incircles (Q2871653)

The following Sangaku problem dates back to 1886: Let \(ABC\) be an equilateral triangle. The side \(AC\) is extended to the point \(B'\), the side \(BA\) is extended to \(C'\), and \(CB\) to \(A'\), such that the triangles \(AB'C'\), \(BC'A'\), \(CA'B'\) and \(ABC\) have congruent incircles. Find the length of the exterior equilateral triangle \(A'B'\) in terms of the length of \(AB\).NEWLINENEWLINEIn this paper this problem is extended to consider regular polygons of arbitrary number of sides. Namely the sides of a regular \(n\)-sided polygon are extended in such a way that the incircles of the new five triangles appearing after the construction are congruent to the incircle of the original polygon. Then, the side of the bigger \(n\)-sided polygon is computed in terms of the side of the original one.NEWLINENEWLINEIt is a nice generalization, the proof is elementary and the paper is worth reading even in the undergraduate level.