If ABC is an equilateral triangle, why would the arc BC be smaller than the arcs AB or AC? I would think that all the arcs would be equal. What the hey? What am I missing here?

Two points (B & C) on a circle delimit two arcs, not three (you are splitting the longer arc). In this case, the short one is B-P-C; the long one goes B-A-C.

If I was to construct an equilateral triangle, then construct each side's perpendicular bisector, then construct the circumcircle, I would had an equalateral triangle inside the circumcircle.

In this case, all vertices of the triangle ABC would be points on the circumcircle. Arc AB = BC = CA since the triangle is equilateral.

Now there would be two paths of "travel" from B to C, one of which would pass through point A. That is, one path would go directly to C (the smaller arc) and the other path would go from B to A then to C (the larger arc).

Is this what you mean by BC delimiting two arcs; one small and one large?

Forget the triangle. Two points on a circle divide it into two arcs. Unless the two points are 180 degrees apart, one of the arcs is smaller than the other. THAT'S the one they are talking about.

I will confess I was taken aback by that myself. Normally when you say "the arc BC", the smaller of the two arcs is what is meant. I suspect this was a case of confusing by trying to be too precise.

The "smaller arc" in this case is precisely what you think of as "the arc BC".