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 $AB$. In 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. It is a nice generalization, the proof is elementary and the paper is worth reading even in the undergraduate level.

Reviewer:

Antonio M. Oller (Zaragoza)