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

March 2025

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 \(\angle AOB\) cannot equal \(60\degree\), 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 \(L_1,L_2,\ldots,L_n\). For each \(i\), label the starting point of \(L_i\) by \(A_i\). Note that the points \(A_1,A_2,\ldots,A_n\) 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 \(n = 7\).

    Before translation

    A regular heptagon with vertices A 1 through A 7, labelled in order in the clockwise direction. Arrow heads indicate that side L 1 points from A 1 to A 2, side L 2 points from A 2 to A 3, and so on.

    After translation

    The seven sides of the heptagon starting at a common centre point and pointing outward forming a shape like an asterisk. B 1 is the outer endpoint of side L 1, B 2 is the outer endpoint of side L 2, and so on.

    1. Show that the polygon \(B_1B_2\cdots B_n\) is a regular \(n\)-gon.

    2. Let \(y\) be the length of \(L_1\) and \(x\) be the length of the line segment \(B_1B_2\). 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 \(n = 4\).