MathLabs

Problem 5

Let nn be an integer greater than or equal to 33. Prove that there is a set of nn points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
Step 5 of 5: Lattice vertices give rational area
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In plain words

Integer coordinates make the shoelace sum integral; dividing by two can only produce a rational number.

2[ABC]=∣det⁡(x1x121x2x221x3x321)∣∈Z2[ABC]=\left|\det\begin{pmatrix}x_1&x_1^2&1\\x_2&x_2^2&1\\x_3&x_3^2&1\end{pmatrix}\right|\in\mathbb Z
Detailed analysis

For any three points (xj,xj2)(x_j,x_j^2), all entries in the shoelace determinant are integers. Thus twice the signed area is an integer, so the area is a half-integer and therefore rational. Together with Step 4, every triple determines a non-degenerate triangle of rational area.