\documentstyle{amsppt} \magnification=\magstephalf  
\hsize=15.8true cm 
\nologo  \TagsOnRight \NoRunningHeads \parskip=4pt \def\a{\angle}

\topmatter\title A Note on Reyes's Theorem about Triangles\endtitle 
\author Don Zagier\endauthor\endtopmatter
In the last issue of {\it Nieuw Archief}, the following elegant
theorem was stated by Reyes [1]:
\proclaim{Theorem} Let $A_1A_2A_3$ be a triangle.  We extend the numbering
cyclically by $A_{i+3}=A_i$ for $i\in\Bbb Z$. Suppose given for each $i\in\Bbb Z$
a point $X_i$ on the perpendicular bisector of $A_{i-1}A_i$ such that the
three points $X_i$, $A_i$, $X_{i+1}$ are collinear for each $i$.  Then
the sequence $\{X_i\}$ has period~6.\endproclaim
\noindent(In the notation of [1] the three vertices $A_1$, $A_2$, $A_3$ were 
denoted by $A$, $B$, $C$ and the first four points $X_1$, $X_2$, $X_3$ and $X_4$ 
by $X$, $Y$, $Z$ and $X'$, and the result was stated in the form that the map
$X\mapsto X'$ from the perpendicular bisector of $CA$ to itself is an involution.)
The proof in~[1], though described as ``quite simple,"  is fairly 
intricate and uses arguments from both vector algebra and
trigonometry.  Here is a much shorter geometric proof.  Let $O$ be the meeting
point of the perpendicular bisectors of the sides (= center of the circumscribed
circle) of the triangle $A_1A_2A_3$.  The point $X_i$ determines, and is
determined by, the angle $\theta_i=\a OA_{i-1}X_i$.  Since $X_i$ is on the
bisector of the angle $A_{i-1}OA_i$, we have $\a OA_iX_i=\theta_i$, and
since $X_i$, $A_i$ and $X_{i+1}$ are collinear we have $\a OA_iX_{i+1}=
\pi-\a OA_iX_i$ or $\theta_{i+1}=\pi-\theta_i$.  Thus the angles $\theta_i$
alternate between two values $\theta_1$ and $\pi-\theta_1$, so $\theta_{i+6}=
\theta_i$ and $X_{i+6}=X_i$.
\bigskip\noindent [1] W. Reyes, 1996, On a theorem in circle geometry, {\it 
Nieuw-Arch.-Wisk\.} {\bf 14}, no\. {\bf 2}, pp\. 231--233.

\end

