MathLabs

Problem 4

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?
Step 1 of 4: Choose the circle construction
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cos⁡θ=35,sin⁡θ=45,Pn=(cos⁡(2nθ),sin⁡(2nθ))\cos\theta=\frac35,\qquad \sin\theta=\frac45,\qquad P_n=(\cos(2n\theta),\sin(2n\theta))
Detailed analysis

Take 0<θ<π/20<\theta<\pi/2 with cos⁡θ=35\cos\theta=\frac35, so sin⁡θ=45\sin\theta=\frac45. For each integer nn, put Pn=(cos⁡(2nθ),sin⁡(2nθ))P_n=(\cos(2n\theta),\sin(2n\theta)) on the unit circle.