Another mathematical puzzle: Why does this work?

Never having bothered to acquire any halfway decent graphics software, I've been using this trick for years to draw regular hexagons, equilateral triangles, and such.

Create a 4-by-4 grid of squares and inscribe a circle in it. The lines of the squares will intersect the circle at 12 points, which appear to be equally spaced at 30-degree intervals around the circumference of the circle. You can then connect selected points to create a triangle, hexagon, or dodecagon.

(The ancient geometers used to take it as a challenge to construct various polygons using only a square and compass. I suppose the modern equivalent would be: Using only a crappy graphics program like MS Paint, construct . . . .)

I can't remember how I happened to discover this -- just one of those happy accidents, I think. I also can't be sure, absent any mathematical proof, that the points are really evenly spaced at 30-degree intervals. (If not, they are certainly close enough that the trick can still be used to create polygons that seem to the human eye to be perfectly regular.) However, I strongly suspect that they are perfectly regularly spaced, because that's just how math is.

The last mathematical proof I embarked on, regarding congruence patterns in triangular numbers (qv) was a matter of straightforward number theory, requiring only everyday algebra, which is why I was never really in any doubt as to my ability to solve it. I am much less confident of my competence in this case, though, being innocent even of basic trigonometry. (Obviously somebody invented trigonometry without having been taught it, and I'm sure I could do the same if push came to shove; the question is whether or not I want to invest so much time and thought in a problem which has elicited only my passing curiosity.)