The definition of a circle is:
Set of points in a plane that are a given distance (Radius) from a given point (Center of the Circle)
Recently I've been hearing some discussions in the usage of π over τ (τ=2π).
π is defined as the ratio of a circle's circumference C to its diameter d:
$$ π = \frac{C}{d} $$
Some people say that if we define the circle using the concept of radius, we should define the circle's constant using its radius as well, doing that we obtain τ:
$$ τ = \frac{C}{r} $$
So, I've been thinking, using that argument, if we want to stick to the definition of π, is it possible to give a definition of the circle without making any reference either explicitly or implicitly to its radius? only using its diameter?
I was thinking in something like this:
Set of points in the plane that belong to a set C, being x an element of C, so that ∀ x ∈ C a straight line starting from x is going to lead to another point in C, being the maximum length of this straight line the diameter of the circle
Could you help me to improve my definition? or maybe give me some ideas of how to fix it?