I propose an alternative: set your compass to BC, draw a circle around A, draw a line from A to any point on the circumference, basta.
Why is his better? We're both using postulate 3. We both end up with the new line at a random angle. But I don't need prop 1 which he uses to draw an equilateral triangle at the outset.
To summarize the discussion in the comments:
Postulate $3$, concerning the construction of circles, is somewhat ambiguous. Modern language agrees with classical in the notion of a center. But modern language would interpret "radius" as a number while, classically, the radius was a line segment connecting the center to any point on the circle.
The proposition in question here resolves the ambiguity. Specifically, it shows that we can use any line segment, to construct a circle with a fixed center. It doesn't matter whether the line segment has the desired center as an endpoint or not.
In the language of construction, the issue is whether or not we work a "collapsing" compass. This was the notion preferred by Plato, and it specifically precludes the ability to "lift the compass off the page" while retaining the measured length. Again, the given proposition shows that whether you prefer rigid compasses or the collapsing sort doesn't change the things you can construct. It just takes more steps to work with the collapsing sort.
this article goes into these (and related) issues with greater depth.