Duplicating the cube: Eratosthenes' mean finder

Cube, Geometry
Eratosthenes solved the problem of duplicating a cube by mechanical means. It was known to the Greeks that the problem of duplicating the cube was equivalent to the problem of finding two mean proportionals between two given line segments of lengths and . The method that Eratosthenes used could be used to find two mean proportionals between any two given segments ( and in the diagram).
  1. Move points and until and have the desired lengths. (Do nothing, or reset the diagram, if you want to find the mean proportionals between 1 and 2.)
  2. Produce the segment .
  3. Drag point until , , and intersect on one point.
  4. Drag point until , , and intersect on one point.
  5. Adjust until these intersections are both determined.
and are the two mean proportionals that we were looking for. Prove it.