You can get right angles with a straightedge and compass, both of which you can make on a construction site with string and pegs in the ground. It's just an instance of bisecting the angle. Which is more convenient in practice is situational.

Right. The big problem with accurately bisecting a 180 degree angle is you need to have access to points rather far in both directions from both points. If you want an accurate right-angle corner near the edge of a property that's flanked on two sides by fences, rivers, busy roads, etc., then you might not have convenient access to one of the anchor points needed to perform your bisection. (Or semi trucks snagging your rope might be inconvenient.)

The nice thing about Pythagorean triples for drawing out foundations is that you don't need access to any ground outside the foundation of your building. Being integers, you also don't need any measuring device apart from some rope. You just pace out a bit under 1/3 of the shortest side (or a bit under 1/5 the longest side, whichever is shorter) (call this an "'bout-right") length of rope. You then use your 'bout-right to make a 3'bout-right, a 4'bout-right, and a 5'bout-right piece of rope. Pull the three ropes tight in your perimeter, and you've got your right-angle for your foundation.

Fair, I didn't realize they were using the string as a compass.

You can certainly get close enough with straight lines. If you create a parallelogram, you can get it square by making the diagonals the same length.

And if you can determine that the diagonals are the same length, you have what you need to get close enough to a parallelogram in the first place.

IIRC, straight lines alone will give you projective geometry (which also has points, but they're the meet of two lines).

> If you create a parallelogram

Parallelism requires affine geometry, which you can't get just with straight lines (and their meeting points). Here are couple of explanations:

- We can also get projective geometry by using great-circles on the surface of a sphere (e.g. "equators" at different angles around the Earth), instead of straight lines on a flat plane: both situations give rise to exactly the same theory. Parallelism doesn't exist on the surface of a sphere, since all great-circles will meet at two antipodal points, so projective geometry (which describes great-circles as well as straight lines) cannot be used to construct/ensure/check that two sides of a quadrilateral are parallel.

- Alternatively, consider that projective geometry is invariant to changes in perspective, whilst parallelism is not. For example, we can get two straight lines by tracing over a photo of train tracks. If the photo was taken top-down, then the lines we traced will be parallel; but if the photo was looking along the track then our traced lines will converge (in the photo, they "meet" at the horizon). Projective geometry (and hence straight lines) can't distinguish between these two scenarios, due to this invariance.

> And if you can determine that the diagonals are the same length

If we extend our straight-line setup with some way to determine parallelism, we still wouldn't be able to compare the lengths of the diagonals, since they go in different directions. Projective geometry + parallelism is affine geometry, which can only compare lengths in the same direction. Essentially, parallelism allows us to translate: we can use this to compare two line segments by translating one so they share a common starting point, then seeing whether the other end has landed closer or further than the first line's. The latter comparison only makes sense if all the points end up colinear (i.e. the original segments were parallel, unlike a pair of diagonals).

To compare the diagonals we also need some form of metric, e.g. like the distance between a pair of compasses.