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.

HINT...if $(a,b)$ is the centre of the circle and its radius is $r$ you can set up and solve a system of three simultaneous equations. So for example, for circle $C_1$ you have $$(a+1)^2+b^2=(r+1)^2$$ and likewise for the other two circles.
Of course, quoting Descartes' Theorem will be a short-cut to finding $r$ but you will still need two of these equations to find $(a,b)$
