Sorry, fell into a little philosophical opaqueness there. Platonist is meant to refer to the ancient philosopher Plato. What he may be most famous for is his theory of Forms. The idea is that there are two worlds: the material world which we can perceive directly and another realm that is "intelligible" but not directly perceivable. We can refer to these as the sensible and supersensible realms. The supersensible realm is accessible through thought and abstraction: it is a world of perfection. It is this realm where perfect circles and shapes and ideas lie. In the Republic, he uses the allegory of the cave to describe the relationship between the supersensible realm and the sensible realm. In a sense, the sensible realm can be imagined as the shadows of figures on a cave wall, where we are unable to turn around and see the figures themselves and only to perceive their projections into the sensible realm. The most modern reinterpretation of Plato's philosophy is present in Judeochristian philosophy. Many would go so far as to say that Christianity is simply a dressed-up reinterpretation of Platonism.
My second section was meant to deal with a proof of God's existence known as the "Ontological Argument." I'm going to reproduce it hear and quote liberally from Wikipeda (I'm kind of lazy):
1. If I am thinking of the Greatest Being Thinkable, then I can think of no being greater
1a. If it is false that I can think of no being greater, it is false I am thinking of the Greatest Being Thinkable
2. Being is greater than not being
3. If the being I am thinking of does not exist, then it is false that I can think of no being greater.
4. If the being I am thinking of does not exist, then it is false that I am thinking of the Greatest Being Thinkable
Conclusion: If I am thinking of the Greatest Being Thinkable, then I am thinking of a being that exists
I know that you are not attempting to prove the existence of God, but the argument bares some similarity, simply replace "Greatest Being" with "Perfect Circle." If there is no circle more perfect then, by the same argument, such a circle must exist. The trouble is this argument is deceptively simple. Bertrand Russell notably said, "The argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies."
Perhaps my favorite refutation comes from David Hume:
[T]here is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.
In this way we can see that conception (of a form or of God or of a mathematical concept) does not lead directly to any statements of existence or non-existence: they simply "are."