Given three segments of length 1, a and b in the plane, how can one construct segments of length a+b, |a-b|, ab, a/b, √a using ruler and compass? Which other calculator functions can be performed by geometric construction?

The first two are trivial. To construct 1/a, draw a triangle with the first side=a and the second side=1. Now construct a similar triangle, but with its first side=1, and its second side will come out to 1/a.

To construct ab, draw a triangle with the first side=1/a and the second side=b. Now construct a similar triangle, but with its first side=1, and the second side will be ab.

To construct a/b, draw a triangle with the first side=b, and the second side=a. Now construct a similar triangle, but with its first side=1, and the second side will be a/b.

Finally, to get √a, draw a semicircle with diameter 1+a. Draw a perpendicular at the point at distance 1 from one extreme of the diameter, and let it intersect the semicircle. The length of that segment will be √a,.