Problem including three circles which touch each other externally The circles $C_{1},C_{2},C_{3}$ with radii $1,2,3$ respectively,touch each other externally. The centres of $C_{1}$ and $C_{2}$ lie on the x-axis ,while $C_{3}$ touches them from the top. Find the ordinate of the centre of the circle that lies in the region enclosed by the circles $C_{1},C_{2},C_{3}$ and touches all of them.
Okay, I can see the lines joining the centres of the circle form a right angled triangle with sides $3,4,5$. But, I can't prosper furthur..any hint or solution.

The expressions are bit complicated, I'll state some facts of outer and inner Soddy circles without proofs.
Denote the corresponding quantities for outer and inner with upper and lower case respectively.
\begin{align*}
 \Delta &= \sqrt{abc(a+b+c)} \\
 \text{radius:} \quad r_{\pm} &= \frac{abc}{2\Delta \mp (bc+ca+ab)} \\
 \text{centre:} \quad S_{\pm} &=
 \frac
   {\left( \frac{\Delta}{a} \mp b \mp c \right) A+
    \left( \frac{\Delta}{b} \mp c \mp a \right) B+
    \left( \frac{\Delta}{c} \mp a \mp b \right) C
   }
   {\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \Delta
    \mp 2(a+b+c)}
\end{align*}
You can get the necessary information by substituting $a=1$, $b=2$, $c=3$, $A=(-1,0)$, $B=(2,0)$ and $C=(-1,4)$ as shown in the figure.

