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Euclidean Geometry Intersection of Circles

I always thought I was good at geometry (at least decent) until I saw this problem today. If anyone could help that would be awesome! I've spent $3 1/2$ hours on this single problem today!

We have points $A$ and $B$ on the lines $y=3$ and $y=12$, respectively. There are two circles that meet the following criteria:

1) They both pass through the points $A$ and $B$ 2) Both are tangent to the x axis at points $P$ and $Q$

If we are given that $PQ=2012$ how would we determine the distance $AB$ ? I would very much appreciate a solution!

thanks!

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    $9$ and $2012$ are really out of scale ; my biggest problem is finding such circles unless $A$ and $B$ are right above one another (i.e. have the same $x$ coordinate). But well, the question is not to draw 2 such circles now is it?2012-02-28
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    Why is the problem repeated twice in the question?2012-02-28
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    You tell me, i dont know2012-02-28
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    Lol wow sorry i was being stupid i'll edit that2012-02-28
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    @Alex Kim: I wrote out a solution to the problem during a boring meeting. Will type it as an answer to your original question, not this one.2012-02-28
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    @AndréNicolas I look forward to seeing your solution. I tried from a semialgebraic-geometry perspective, but the system of equations was too large to manage nicely.2012-02-28
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    @Alex Becker: Done, except for typo hunting. The equations were very nice, if handled the right way. Actually, there is an even nicer way, but I resisted the temptation.2012-02-28
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    This is a repeated question http://math.stackexchange.com/questions/113344/euclidean-geometry-intersection-of-circles Moderators please delete2012-02-28

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