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June 22nd, 2012, 10:05 AM  #11  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Rational Points on a circle Quote:
 
June 22nd, 2012, 10:08 AM  #12  
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Rational Points on a circle Quote:
 
June 22nd, 2012, 10:14 AM  #13 
Member Joined: Oct 2011 Posts: 81 Thanks: 0  Re: Rational Points on a circle
Let's assume for eg. there are only two rational points on the circle, so the chord passing through them will have a rational slope and perpendicular bisector from centre will also have rational slope. But in reality, perpendicular bisector has irrational slope, that is a contradiction, so (assumption was wrong) even two points on a circle can't be rational.(right?)

June 22nd, 2012, 10:31 AM  #14  
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Rational Points on a circle Quote:
I have rather a lot of caveats with the wording and am making the odd assumption when reading that. But to go into them would deflect from the main point of the post; and actually serve no purpose. An elementary introduction to logic, or propositional calculus, would be a useful read, and cover all you need I think, if you feel so inclined.  
June 22nd, 2012, 11:32 AM  #15 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Rational Points on a circle
let be a rational point, that is Make a circle with center passing this point, and assume it should pass some other rational point: We know, Rearrange (1), we obtain: We know the truth that: So the left of (2) must be rational, but the factor in the right should be irrational. In order to make the right of (2) be a rational (and make left=right), you should let , that will result in right=0 and left =0 too. Therefore or , or the circle passes two rational points at most: and . 
June 22nd, 2012, 12:51 PM  #16 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Rational Points on a circle
Q2 It is true while the fixed line is the perpendicular bisector to some segment, because any point on the fixed line has equidistance from both endpoints of the segment. When one end point of the segment is a fixed point, the other end point is the other fixed point in this case. 
June 22nd, 2012, 01:25 PM  #17  
Member Joined: Jun 2012 From: UK Posts: 39 Thanks: 0  Re: Rational Points on a circle Quote:
 
June 22nd, 2012, 01:37 PM  #18  
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Rational Points on a circle
Simple proof bout this: Quote:
Now define another rational number . , . So a) , since , . b) , since , . c)Suppose , if , . , . But according to a), we know, , that is contradiction of . So too. d) Similar as c), let , , if ,  
June 22nd, 2012, 01:38 PM  #19 
Senior Member Joined: Oct 2010 From: Changchun, China Posts: 492 Thanks: 14  Re: Rational Points on a circle
Thanks, ghostwalker!


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