
06243736
j
2014d.00627
Ito, Naoharu
Wimmer, Harald K.
A Sangakutype problem with regular polygons, triangles, and congruent incircles.
Forum Geom. 13, 185190 (2013).
2013
Florida Atlantic University, Department of Mathematical Sciences, Boca Raton, FL
EN
G45
Sangaku
incircle
regular polygon
http://forumgeom.fau.edu/FG2013volume13/FG201319index.html
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.
Antonio M. Oller (Zaragoza)