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 2 of 5: Factor the distance between two points
Previous stepNext step
In plain words

The difference of squares extracts an integer factor and leaves a square root whose radicand is just one more than a square.

d=(x1−x2)2+(x12−x22)2=∣x1−x2∣1+(x1+x2)2d=\sqrt{(x_1-x_2)^2+(x_1^2-x_2^2)^2}=|x_1-x_2|\sqrt{1+(x_1+x_2)^2}
Detailed analysis

For distinct points (x1,x12)(x_1,x_1^2) and (x2,x22)(x_2,x_2^2), the distance formula and x12−x22=(x1−x2)(x1+x2)x_1^2-x_2^2=(x_1-x_2)(x_1+x_2) give d=(x1−x2)2+(x12−x22)2=∣x1−x2∣1+(x1+x2)2d=\sqrt{(x_1-x_2)^2+(x_1^2-x_2^2)^2}=|x_1-x_2|\sqrt{1+(x_1+x_2)^2}.