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Show 2 of 3 points on a circle are the diameter 
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#1
Nov2212, 03:31 PM

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1. The problem statement, all variables and given/known data
Points A B and C lie on the circumference of a circle where $$A =(3,2)\\B=(1,6)\\C=(7,2)$$ Show that AC is the diameter of the circle. 2. Relevant equations 3. The attempt at a solution Would it be sufficient to show that the angle ABC is a right angle and therefore by the inscribed angle theorem that any angle subtended from a diameter is a right angle? 


#2
Nov2212, 04:04 PM

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I think you can finish from there. 


#3
Nov2212, 04:13 PM

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So is the diameter not the only 2 points you can subtend right angle triangle from then? (I can't picture any other cases) 


#4
Nov2212, 04:58 PM

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Show 2 of 3 points on a circle are the diameter



#5
Nov2212, 05:00 PM

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#6
Nov2212, 05:15 PM

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Coordinate geometry for finding equation for the circle which contains those three points could be easier to handle. The resulting and simplified equation will show the size of the radius; then, you merely use distance formula to show that the intended given points (giving length, AC) are twice the size of radius.



#8
Nov2212, 09:34 PM

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My earlier posted method of coordinate geometry to try to make a system of three equations would not be very neat to solve. 


#9
Nov2212, 09:39 PM

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EDIT: Just having solved this most of the way through, what I suggest is very good; but not necessary to check the slopes of any segments. You can if you wish. IF longest side is diameter, then its midpoint is the center of the circle. Longest side length indicates the diameter from which you get the size of radius. You use the found center point of this longest side to fillin the standard form of equation for a circle. NOW, you can check to see if each point satisfies the equation. 


#10
Nov2212, 09:54 PM

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I'm not sure why everyone has been leading the OP away from his solution. Showing angle ABC to be a right angle is an ingenious solution, considering it's very simple to do and it's short and sweet.



#11
Nov2212, 10:44 PM

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The logic of that last part is backwards from what's needed here, as in "A implies B" versus "B implies A". It just has to be changed to "any chord that subtends a right angle at the circumference is a diameter" (that is, assuming that the 'inscribed angle theorem' says that). 


#12
Nov2212, 11:18 PM

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Oh, and it's not your part I was particularly concerned with But I realize now it was a bad choice of words to say that "everyone" has been leading the OP away... 


#13
Nov2312, 01:16 PM

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Thanks,
I think I maybe got the circle theorem wrong in my OP http://en.wikipedia.org/wiki/Thales%27_theorem Is that basically what I was talking about showing ABC to be right angled would be sufficient to show the AC is the diameter? 


#14
Nov2312, 03:56 PM

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