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Constructible Points and Numbers

In order to analyse the construction power of straightedge and compass, we need to use coordinate geometry.
Constructible points are simply points that can be produced by Euclidean constructions. A more rigorous definition of Euclidean construction is needed: 
  1. We should get rid of any randomness - we are not allowed to select an "arbitrary point" in the process of construction.  Any new point should come from an intersection between lines or circles that are drawn previously.
  2. Obviously, we need to have at least two given points to start a construction.  By convention,  and and the two initially given points.
 is called a constructible number if  is a constructible point.  We have the following useful result:

is a constructible point if and only if and are constructible numbers.

Exercise In the following applet, construct the point  from the points  and using straightedge and Euclidean compass and hence show that it is a constructible point. Note: No "Well done!" message will appear for this exercise.