Two different sized circles lie on an line, touching each other at a point. A small circle is inscribed in the space between. How does its radius depend on the radii of the two larger circles?

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    $\begingroup$ I solved your problem. If you want to see my solution, show us your attempts. $\endgroup$ – Michael Rozenberg Feb 25 at 19:32

We have 3 triangles such that each are tangent to each other and each are tangent to some line. Lets say the radius a,b are given, and we must find c.

enter image description here

We can use the Pythagorean theorem to find the lengths $x, y$ and $x+y$ in terms of $a,b,c$

Then $x + y = x+y$ should give you enough information to solve for $c$ in terms of $a,b.$


Let $k_n$ be the curvature (i.e. $=r_n^{-1})$ of the $n$th circle

In virtue of Descartes' theorem, the curvature $k_4$ from the radius of the smaller circle is $$k_4=k_1+k_2+k_3\pm2\sqrt{k_1k_2+k_2k_3+k_1k_3}$$

Since one of the 'circles' is a line, its curvature is $0$. Thus


The $\pm$ symbol depends on how the fourth circle is tangent to the others: enter image description here

Since you mentioned that the fourth circle would be 'a small circle', I guess your formula turns into


Once obtained the value of $k_4$, simply use the fact that $$k_4=\frac{1}{r_4}\iff r_4=\frac{1}{k_4}$$


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