Problem 5

Let

n

be an integer greater than or equal to

3

. Prove that there is a set of

n

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 3 of 5: The distance is irrational

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In plain words

There is no integer square between two consecutive integer squares, so adding one under the radical creates the required irrationality.

(x_1+x_2)^2<1+(x_1+x_2)^2<(x_1+x_2+1)^2

Detailed analysis

The integer $1+(x_1+x_2)^2$ lies strictly between the consecutive squares $(x_1+x_2)^2$ and $(x_1+x_2+1)^2$ , so it is not a perfect square. Its square root is irrational. Since $|x_1-x_2|$ is a nonzero integer, multiplying by it preserves irrationality; hence every pairwise distance is irrational.