
Journal for Geometry and Graphics 11 (2007), No. 2, 165171 Copyright Heldermann Verlag 2007 Two Kinds of Golden Triangles, Generalized to Match Continued Fractions Clark Kimberling Dept. of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722, U.S.A. ck6@evansville.edu [Abstractpdf] Two kinds of partitioning of a triangle $ABC$ are considered: sidepartitioning and anglepartitioning. Let $a = BC$ and $b = AC$, and assume that $0< b \leq a$. Sidepartitioning occurs in stages. At each stage, a certain maximal number $q_n$ of subtriangles of $ABC$ are removed. The sequence $(q_n)$ is the continued fraction of $a/b$, and if $q_n=1$ for all $n$, then $ABC$ is called a sidegolden triangle. In a similar way, anglepartitioning matches the continued fraction of the ratio $C/B$ of angles, and if $q_n=1$ for all $n$, then $ABC$ is called a anglegolden triangle. It is proved that there is a unique triangle that is both sidegolden and anglegolden. Keywords: Golden triangle, golden ratio, continued fraction. MSC: 51M04 [ Fulltextpdf (131 KB)] for subscribers only. 