Regula Hexagon

[RonPurewal]

I made this assumption while doing the problem, but I was wondering how we know that Circle O is the same size as the other circles? Is there a math rule regarding tangent circles?

If you draw the "diameters" of the hexagon, you'll get six triangles (ABO, BCO, CDO, and so on).
These are equilateral triangles. There are ways to prove that fact, but it's easier just to realize that the hexagon is symmetrical.

Now, just think about any of these triangles.
If the middle circle had any other radius, then they'd be non-equilateral triangles. (Say the middle radius is "r" and the other ones are "R". Then, looking at triangle ABO, you'd have side AB = 2R, but side AO = side BO = R + r. That's only going to be equilateral if R and r are the same.)

[RonPurewal]

Also -- if you have issues like this when you encounter a problem, just try drawing the situation.

I.e., draw the six outer circles. They have a fixed size, since the radius of each is 1/2 the side of the hexagon.
Once those are there, it should be pretty clear that the entire structure has a fixed size and shape -- meaning that there's only one possible size of the inner circle. (If you draw the inner circle any smaller, there will be gaps. If you draw it any bigger, it will overlap the outer circles. Try it.)