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Journal for Geometry and Graphics 11 (2007), No. 2, 165--171
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.


Two kinds of partitioning of a triangle $ABC$ are considered: side-partitioning and angle-partitioning. Let $a = |BC|$ and $b = |AC|$, and assume that $0< b \leq a$. Side-partitioning 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 side-golden triangle. In a similar way, angle-partitioning matches the continued fraction of the ratio $C/B$ of angles, and if $q_n=1$ for all $n$, then $ABC$ is called a angle-golden triangle. It is proved that there is a unique triangle that is both side-golden and angle-golden.

Keywords: Golden triangle, golden ratio, continued fraction.

MSC: 51M04

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