Solve for possible radii of fourth circle given three circles.

Question

Question: The centers of three circles, each with a radius R, are located on a straight line at points A, B and C. It is known that AB=BC=3. A fourth circle touches each of the three circles. Find all possible radii of the fourth circle if

a) R = 1 b) R = 2 c) R = 5

I am a bit confused as to how to begin solving the problem, as this is a trignometry question but I do not know where I can relate the triangle into the problem.

Answer 1

We just need to use Pythagoras' Theorem.

When $R=1$, there are two possible cases, with the radius equals to $\frac{9}{4}$ and $\frac{9}{2}$ respectively.

When $R=2$, there is one case, with the radius equals to $\frac{9}{8}$.

When $R=5$, there are two cases, with the radius equals to $\frac{9}{20}$ and $\frac{9}{10}$ respectively.

Answer 2

Here's a hint to get you started (this is for the case $r=1$). Discovering how to draw a fourth circle tangent to the first three is crucial before setting up any equations.