In order to analyse the construction power of straightedge and compass, we need to use coordinate geometry.[br][br]

[b][color=#0000ff]Constructible points[/color][/b] are simply points that can be produced by Euclidean constructions.[br][br]A more rigorous definition of Euclidean construction is needed: [br][br][list=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.[/*][*]Obviously, we need to have at least two given points to start a construction.  By convention, [math]\left(0,0\right)[/math] and [math]\left(1,0\right)[/math] and the two initially given points.[br][/*][/list][br][math]x[/math] is called a [b][color=#0000ff]constructible number[/color][/b] if [math]\left(x,0\right)[/math] is a constructible point.  We have the following useful result:[br][br][i][b][center][math]\left(x,y\right) [/math] is a constructible point if and only if [math]x[/math] and [math]y[/math] are constructible numbers.[/center][br][br][/b][/i][b][br][br]Exercise[/b][br][br]In the following applet, construct the point [math]\left(\sqrt{2},\sqrt{3}\right)[/math] from the points [math]\left(0,0\right)[/math] and [math]\left(1,0\right)[/math] using straightedge and Euclidean compass and hence show that it is a constructible point.[br][br][b]Note[/b]: No "Well done!" message will appear for this exercise.