I honestly do not know why I'm so lost here, but I am. Really, I'm proving the construction of an equilateral triangle inscribed in a circle, but the inscribed regular hexagon is crucial to that proof (at least the way I'm proving). There are proofs online, but they are not satisfying to me. Here is my question. So, we construct a circle, then label any point P on the circle, then we swing 6 arcs across the circumference of the circle starting at point P and end on our original point while keeping the compass set to the length of the radius of the circle. Does this process not imply that the circumference of a circle is 6 times the radius? We know that this is not true... what am I missing here?
No, it does not mean anything about the circumference of the circle. You are not measuring the circumference with your compass.
You just measure the cord not the the length of the arc.