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I saw the statement in the question title online. It doesn't seem hard to show this is true simply by manipulating the expression $\frac{abs(p-q_1)}{abs(p-q_2)}=d$ (though I haven't done this), but there has to be some geometric explanation I'm missing. I can, geometrically, see why the set describes a line when $d=1$ (we construct a triangle with edges $p$, $q_1$, $q_2$ and look at the median), but why is it a circle otherwise?

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This phenomenon is called Apollonius Circle. It has a nice projective geometry solution, and of course, a brute force coordinate geometry solution. – Calvin Lin Jan 9 '13 at 23:50

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