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A theorem states, that the three Apollonian circles, associated with the given triangle $ABC$ with sidelengths $a \neq b \neq c$ intersect in two points. The proof proceeds by showing that if the two circles passing through $A$ and $B$ intersect at some point $P$, then the third one also passes through $P$. But how does one prove that the former two circles intersect in the first place?

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My dear, its an assumption ( I think so, may be or may not be, the other MO users will step within a few time and help you in detailed manner ) . First you need to assume that they intersect at some point and then proceed further. In order to prove something, you need to assume something. BTW, its good that you mention the theorem name and link if possible. –  Iyengar Aug 13 '12 at 8:47
    
If you're allowed the use of coordinate geometry, this should not be too hard... –  J. M. Aug 13 '12 at 11:26
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