I was helping a friend out with an SAT problem, when an odd thought popped up in my head. Why is the center of the diameter of a circle the actual center of the circle? How could I prove this? Is this just the way a circle is defined?
Great question - a fact which is "obvious" but takes a bit of thought to see for sure why it's true.
The main thing is to be absolutely clear about what terms mean.
Definition. A circle (in a given plane) is the set of all points at a given (positive) distance from a given point. The given point is called the centre of the circle. A diameter of the circle is a line segment passing through the centre and having its endpoints on the circle.
Note that the definition of "diameter" doesn't say that the centre of the circle is its midpoint. It just takes a little more work to prove this.
So, consider a diameter $AB$. By definition the centre $C$ is somewhere on $AB$. Since $A$ and $B$ are on the circle, the definition says that $AC=BC$. Hence $C$ is the midpoint of $AB$.
Comment. Euclid's definition (from proofwiki):
a diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.
So it looks as if Euclid just assumed the centre of the diameter is the centre of the circle, and did not even think to ask the question you have asked!!!!