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A Sangaku-type problem with regular polygons, triangles, and congruent incircles. (English)
Forum Geom. 13, 185-190 (2013).
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)
Classification: G45