Where will the ants position themselves so that they are precisely twice as far from vinegar as they are from peanut butter?

Question

An ant will always position itself so that it’s precisely twice as far from vinegar as from peanut butter. If we put a dab of vinegar at A and a dab of peanut butter at B and we release a troop of ants, what formation will they take up?

Note: an ant won't position itself on top of another ant.

Source: futilitycloset.com

P.S. Please don't just state the answer. Please provide proof why your answer will work.

Answer 1

The answer is

A circle

Solution

From Circle of Apollonius, we can see that the locus of points that maintain a constant distance ratio to two points is a circle. In fact, the circle with diameter $CD$, where $C$ is the point on segment $AB$ such that $AC:BC=2:1$ and $D$ is the point on line $AB$ but outside segment $AB$ such that $AD:BD=2:1$, is our desired circle.