Construction problems in geometry II

First of all I would like to share with you a book that I got yesterday (it was a real surprise from Amazon, the book was delivered at 8:00 PM, and I expected it only today or tomorrow).  The secrets of triangles is a great book.  It includes, among others, a big section on constructing triangles using a compass and straight edge given three different elements of a triangle.  I started to read it and I am really enjoying this reading.

Now back to problems.  A separate group of construction problems consists of ones that should be solved by a compass only.  My favorite one was introduced by Napoleon (yes, the emperor).
Given a segment of the length one, construct a segment of the length $\sqrt{2}$ units.
Obviously, you cannot draw a segment without a straight edge, just find the endpoints.

The solution to the problem from yesterday's post is given by two figures below.

AB is a given diameter and C is a given point.  AC intercepts given circle at D, and BC intercepts it at E.  Angles ADB and AEB are right angles because AB is a diameter.  AE and BD are altitudes of BC and AC respectively.  The line from C through the point their intersection is an altitude to AB.