There's a formula for the coördinates of where two arcs meet, where a line and a circle meet, etc., in terms of the coördinates of the points which generated th

There's a formula for the coördinates of where two arcs meet, where a line and a circle meet, etc., in terms of the coördinates of the points which generated the lines and circles — and you only need a few such formulæ (linear and quadratic equations, in fact) to cover all legal straight-edge-and-compass constructions. So keeping track of exact, algebraic locations of the points is basically equivalent to manipulating exact expressions for roots of polynomials, which is not totally trivial (the tricky bit is simplifying roots of a quadratic equation whose coefficients were given as roots of a previous quadratic equation) but it's the sort of thing Maple and Mathematica have been doing for decades.