| | That follows from the axioms and definitions of geometry. You don't prove similarity with pi, it's the other way around: from the similarity of all circles we know that the ratio circumference to diameter of all circles is the same and therefore a fixed number; that number we call pi. If all circles were not similar, pi wouldn't be a fixed constant. Trying to prove similarity by using pi would be begging the question.
Well at the risk of sounding too pedantic, all circles, by the definition of a circle, must be similar. How we define a circle is how we define the relationship between geometric qualities in that shape. So I would interpret that to mean we have come up with the concept of a geometric shape called "circle" to mean it has a diameter, and who's circumference is determined by multiplying it by Pi. You say if all circles were not similar, that Pi would no longer be a fixed constant, but that would mean we have changed the definition of circle. Otherwise you would be probably be speaking of an oval, or some other shape such as a rhombus or rectangle. All geometric shapes have qualities to them that give them their definition.
A square is defined as a parallelogram, with 4 equal sides, with 4 90 degree angles. We can expound on that further to explain what we mean by sides and what we mean by angles and what we mean by parallelogram, but in all of those instances, we are required to use a definition as part of that explanation. We can keep defining words until we strip them down to some a priori knowledge. Ultimately, all language is comprised of words that have a definition to them, that gives them some coherent meaning to them when two people converse with each other.
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