After constructing the two rays at an angle of approximately 60 degrees, the author goes about constructing an actual (approximate) equilateral triangle by drawing two perpendiculars. That might be useful if one needs a triangle of precisely that size, but given a pair of compasses (or a straightedge with two markings), one gets any ≈equilateral triangle by just marking the two intersection points of any circle around the origin with the two rays. I believe this method doesn't give anything less precise than the method outlined in the post.

Given that the idea was to make the construction simple and small, I'm surprised the compasses method wasn't chosen.