Can we always draw a circle that is internally tangent to three circles? Here is the picture of the problem:

I'm looking for an intuitive explanation and also an outline of how a rigorous proof would look like.
Our professor directly started to speak about finding the radius of the small circle, but it got me thinking. How can we ensure that such a circle always exists?
I researched and came across Appolonius Problem, but I couldn't find an explanation for my problem. I know that three points define a circle, but how does that help me?
Thank you in advance.


Hint: As can be seen in figure:
1- Connect centers A, B and C of three circles.
2-Find circumcenter D of ABC.
3-E is the center of big circle and F is that of small one. By accurate drawing we can see that D is midpoint of EF. The radius of big circle is:
$R=DA+\frac {AA'+BB'+CC'}3$
And that of small one is:
$r=DA-\frac {AA'+BB'+CC'}3$
4- for big circle take for example point A' and B' as centers of two circles which intersect at E, this is the center o big circle.
5- For small circle take for example two points, intersection of DA and DB with circles A and B, as centers of two circle with radius r which intersect at F , this will be the center of small circle.You may also connect D to E and extent it equal to DE from D to find F.
