Find circumcenter when distance between ABC points of triangle with two points's ratio given

Question

The complete problem is:

I am having three points A,B,C whose ratio of the distances from points (1,0) and (-1,0) is 1:3 each. Then I need the coordinates of the circumcenter of the triangle formed by points A,B & C.

Can anybody tell me how to proceed?

Answer 1

I'd proceed like this:

1. Prove that, in 2D, the set of all points defined by a fixed ratio $\rho\neq1$ of their distances to two distinct given points (which I will call poles for reasons not explained here) is a circle $K$. (I'd use equations with cartesian coordinates for that, but if you find a coordinate-free reasoning, that would be interesting.)
2. Conclude that the circumcenter of $\triangle ABC$ is the (yet unknown) center $M$ of that circle $K$.
3. Find two points $D,E$ on the line through the poles (here: the $x$-axis) with the same given distance ratio $\rho$.
4. Argue with symmetry and conclude that $M$ must be the midpoint of the segment connecting $D$ and $E$. In particular, $M$ lies on the $x$-axis.

Update: An illustration:

More could be said on this ratio-and-circle topic, but I have kept that to the comments section.