You probably need to read about this construction, else it makes no sense. My labeling is adapted from the description given in "Problem studies #4.2" in Howard Eves' An Introduction to the History of Mathematics, 6th ed., pp. 124-125, which I quote below.

Let a and b, a > b, be two given line segments. In a horizontal plane, draw a circle on AD=a as diameter and construct chord AB=b. Let AB produced meet in point P [don't you love that kind of talk?], the tangent to the circle at D. Vertically erect the upper half of a right circular semicylinder on the semicircle ABD as base; generate a right circular cone by rotating AP about line AD; generate a torus of zero inner radius by rotating, about the element of the semicylinder through A, the vertical circle on AD as diameter. Denote by K the point common to the semicylinder, the cone, and the torus. and let I be the foot of the semicircle ABD of the element through K of the semicylinder. Prove that AK and AI are the two mean proportionals between a and b; that is, show that AD:AK=AK:AI=AI:AB.

For several reasons, you will find labels of points K and I below that do not correspond to the description above. One reason is the labels got too crowded, hidden, and/or ambiguous (I now forget how badly) when drawn near those points. Instead, I use corresponding points in the original plane, with labels K and I, such that the lengths AK and AI are clearly unchanged. For example, the point now labeled K in the plane is the result of taking the intersection of the three surfaces (the original point K) and rotating it back down to the plane via the cone. You can locate a green arc that indicates this. For the point I, rotate about the original about the vertical axis through A along a circle in the torus, and then project down to the plane, or vice versa. The arcs are blue, the projections are along the red segments.

The other reason for the procedure described above is that the result shows the relevant lengths in the plane (turn the graphic upside down) and reveals a simple configuration that demonstrates the two mean proportionals. (Apologies if you don't like the color of the text! I couldn't find one I liked.)

Some intelligent history is on the MacTutor History of Mathematics archive and on the Stanford Encyclopedia of Philosophy.

Grab it, turn it, spin it with the mouse.

You need Java installed and enabled to execute this applet. You are missing out on the ability to see and manipulate a figure like this: — RM

This image was produced using Mathematica and converted to the present form with LiveGraphics3D.