Problem of the Month
Problem 6: Regular Polygons and Lattice Points

A lattice point in the Cartesian plane is a point $(a,b)$ with the property that both $a$ and $b$ are integers. In this Problem of the Month, we will investigate regular polygons that have every vertex lying on a lattice point.

1. Let $A$ and $B$ be distinct lattice points on the Cartesian plane, neither of which have coordinates $(0,0)$. Show that the measure of $\mathrm{\angle }AOB$ cannot equal $60\mathrm{°}$, where $O$ has coordinates $(0,0)$.

2. Consider a regular pentagon $ABCDE$. Let $F$ be the point of intersection of lines $AD$ and $BE$.

   

   1. Show that the quadrilateral $FBCD$ is a parallelogram.

   2. Show that if $B$, $C$, and $D$ are lattice points then so is $F$.

3. View a regular $n$-gon as a collection of $n$ line segments. Give each line segment a direction (indicated by an arrow), moving in a clockwise direction (see the image below). Label the line segments $L_1,L_2,\dots ,L_n$. For each $i$, label the starting point of $L_i$ by $A_i$. Note that the points $A_1,A_2,\dots ,A_n$ are the vertices of the $n$-gon.

   Now, translate the line segments (without any rotation) so that the points $A_i$ all coincide. For each $L_i$, label its new endpoint by $B_i$. Below are images of this process when $n=7$.

   

   

   1. Show that the polygon $B_1{B}_2\cdots B_n$ is a regular $n$-gon.

   2. Let $y$ be the length of $L_1$ and $x$ be the length of the line segment $B_1{B}_2$. Compute $\frac{x}{y}$ in terms of $n$.

4. We call a polygon that has every vertex lying on a lattice point a lattice polygon. Show that if a regular $n$-gon is a lattice polygon, then $n=4$.