Verify Conway’s construction for taking cube roots using a compass and a straightedge with a unit distance marked on it.
Suppose we have constructed a distance , with , with endpoints and . Construct the line perpendicular to through , and construct the point on this line whose distance from is . (Note that, by the Pythagorean theorem, the distance from to is 1.) Now construct on such that is between and and the distance from to is . Extend the line through and . Now, using the marked straightedge, construct the line passing through whose points of intersection with and are separated by a distance of 1. Call these points and , and call the distances and and , respectively, as in the following diagram.
Finally, construct the lines through perpendicular to and , and call the distances from to these lines and , respectively, as we have labeled in green in the following diagram.
Now and the triangle with vertices and and a side length are similar, so that (1) . Similarly, and the triangle with vertices and and side length are similar, so that (2) . Likewise, and the triangle having vertices and and a side length are similar, so that (3) .
Finally, using the Pythagorean theorem on , we have (4) . We can expand (4) and simplify to get the equation (4′) .
Solving (1) and (3) for , we see that . Solve for to get (5) . Equating with (2), we have , which simplifies to (6) . Using (6) to eliminate in (4′), we have . (Hint: clear denominators and note that is a factor.) Similarly, eliminating yields , as desired.