Can we colour...

Plato

MHF Helper
Aug 2006
22,506
8,663
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.

View attachment 37405
There does not exist a unit circle that contains three points that are pair-wise one unit from one another.

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 $\dfrac{\sqrt3}{3}$.
 

ChipB

MHF Helper
Jun 2014
312
130
NJ
There does not exist a unit circle that contains three points that are pair-wise one unit from one another.
That's not what the OP asked about. What he said was: "every two points which are distant from each other in the lenght [sic] of 1." He doesn't ask about three points all equidistant from each other.
 
Feb 2017
13
1
Los Angeles, CA
Yeah, and they have to be on a circle.