id: 06243736
dt: j
an: 2014d.00627
au: Ito, Naoharu; Wimmer, Harald K.
ti: A Sangaku-type problem with regular polygons, triangles, and congruent
incircles.
so: Forum Geom. 13, 185-190 (2013).
py: 2013
pu: Florida Atlantic University, Department of Mathematical Sciences, Boca
Raton, FL
la: EN
cc: G45
ut: Sangaku; incircle; regular polygon
ci:
li: http://forumgeom.fau.edu/FG2013volume13/FG201319index.html
ab: 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.
rv: Antonio M. Oller (Zaragoza)