I had $3$ circles of radii $1$, $2$, $3$, all touching each other. A smaller circle was constructed such that it touched all the $3$ circles.
What is the radius of the smaller circle?
This is what I did:
I conveniently positioned the $3$ circles on the coordinate axes and found the coordinates of the centers of each circle.
Then I wrote a general equation of a circle( for the smaller one) , and using the fact that distance between the centers is equal of the sum of radii ( for circles touching each other) I found $3$ equations, which I could use to solve for the variables in the general equation. Hence I found the equation of the smaller circle, and thus its radius.
However, I believe that this method is very inefficient as I ended up with so many steps and sub steps to solve the equations.
Is there a better way to approach this?
I would prefer a geometrical solution instead of a coordinate solution.
Thanks for the help!!
I found that this question already has an answer here:https://math.stackexchange.com/questions/1867315/problem-including-three-circles-which-touch-each-other-externally
However is there a more efficient solution for this than what was mentioned in those answers?