Although it is not immediately obvious, you can derive the geometric mean of x and y, thanks to similar triangles. Draw a horizontal segment ab of length y. Place the point c between a and b, such that ac has length x. Let m be the midpoint of ab and draw a circle, centered at m, whose diameter is ab. Draw the verticle line through c and let it intersect the circle at d. Now ∠ adb is an inscribed angle, and is half the central angle (which happens to be a straight line). Therefore adb forms a right triangle. At the same time, acd forms a right triangle, and both triangles share a common angle at a. They are similar triangles, and the distance ad = sqrt(xy).
Apply this procedure to the distances 1 and y and find the square root of y. Irrational numbers have entered our world! (We saw this earlier when we constructed the square root of 3.) Of course most of the real numbers will always be inaccessible, since they are uncountable, while sequences of construction operations are countable.
Deriving x2 is not hard. Let the horizontal segment ac have length 1. Draw a circle of radius x about a, and draw a vertical line through c. Circle and line intersect in d. Draw a line through d, perpendicular to ad. Let this line intersect the line ac in the point b. Once again triangles acd and adb are similar. The segment ab has length x2.
This only works for x > 1. If x is smaller than 1, it is certainly larger than 1/n for some n. Set the length of ac to 1/n and follow the above procedure. The resulting segment ab has length nx2. Divide this by n to get x2.
Multiplication is an immediate corollary. Take the geometric mean of x and y and square it to get xy.
Now let's find the inverse of x. Reverse the roles of 1 and x in the previous construction. Thus ac has length x, which we assume is less than 1, and a circle of radius 1 is drawn about a. If y is the length of ab, we have x is to 1 as 1 is to y, hence y = 1/x.
It is interesting to see how this construction leaves 1/0 undefined. When x is 0 c and a coincide. The line through d runs parallel to b, and there is no way to complete the diagram.
If x is larger than 1 it is certainly less than n for some n. Draw a circle of radius n, instead of radius 1. This gives: x is to n as n is to y. So y = n2/x. Divide by n2 to find the inverse of x.
The standard mathematical operators have been implemented, and the set of constructible distances forms a field, with the rationals as subfield. If an additional distance is brought in, call it w, all the distances in the field extension Q(w) are accessible. This includes 3+w2, 17/w, etc. Each new distance establishes a larger field extension, building a tower of fields within the reals.