There does not exist a unit circle that contains three points that are pair-wise one unit from one another.Remember the radius of the circle is 1 and the distance between points that we are interested is also 1. For a circle of radius 1 the inscribed equilateral triangle does not have side lengths of 1.

What I tried to explain was an approach of dividing the circle into six sectors, like the attached figure, where any two points that are exactly a distance of 1 away from each other are different colors.

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Any set of three points that are pair-wise one unit from one another form determines an equilateral triangle of sides of length one. It can be circumscribed by a circle with a radius of $\dfrac{\sqrt3}{3}$.