4
$\begingroup$

Given two fixed points A and B, find the locus of the point P, satisfying PA=2PB. Of course we can use Cartesian geometry to find the equation of the curve.

Let the midpoint of A and B be the origin, and the line AB be the x-axis, then the coordinates of A and B is (a,0), (-a,0). Let the coordinates of P be (x,y). Then $\sqrt{(x-a)^2+y^2}=2\sqrt{(x+a)^2+y^2}$, we get $(x+\frac{5}{3})^2+y^2=(\frac{4}{3}a)^2$. So the locus is a circle. But how can we solve this problem by pure geometry?

  • 4
    [Apollonian circles.](https://en.wikipedia.org/wiki/Circle#Circle_of_Apollonius) Have you encountered them?2011-12-08
  • 1
    This is covered in C. Stanley Ogilvy's book _Excursions in Geometry_, a beautiful book that requires nearly no background in math to read it, beyond the simplest parts of algebra and geometry. See the section on pages 14--17 titled "The Circle of Appollonius".2011-12-08
  • 0
    [Here's the main article on Apollonian circles](https://en.wikipedia.org/wiki/Apollonian_circles).2011-12-08

1 Answers 1