You have 2 identical unit circles (radius 1) tangent to each other. On top between them a small circle is placed such that it's tangent to both the unit circles. The blue line is tangent to all three circles.
Here is an image depicting what I mean:
Question: What is the radius of the small circle?
Is there an easy way to solve this problem? I'm sure there is, but I can't seem to come up with a simple method. It looks like it's around 1/4. How would you guys approach this problem or problems of this type in general? Please, try to explain it, if possible, in simple terms.
Best Answer
You may also use the Descartes theorem. If four circles with radii( $r_1, r_2, r_3, r$) are tangents , then we have:
$$(\frac1{r_1}+\frac1{r_2}+\frac1{r_3}+\frac1{r})^2=2(\frac1{r_1^2}+\frac1{r_2^2}+\frac1{r_3^2}+\frac1{r^2})$$
Here $r_1+r_2=1$ and $r_3=\infty$ which represent the line , so we have:
$$(1+1+0+\frac 1 r)^2=2(1+1+0+\frac 1{r^2})$$
which finally gives:
$4r+1=2\Rightarrow r=\frac14$