Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.