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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
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