The circles are tangent two by two. The radius of the largest circle is 2. We have to calculate the radius of the two smaller ones.
I am able to calculate the distance based on the following sketch:
Let $x$ be the unknown distance. Then it follows that: $AM=1+x$, $OM=2-x$, $OA = 1$. From Pythagorean theorem $AM^2 = OM^2 + OA^2$. Substituting and solving for $x$, the radius is found to be $2/3$.
My question is how we can justify geometrically the construction. How M is defined as the center of the small circle and why MO is perpendicular to AO (in order to be allowed to apply Pythagoras' theorem).
Instructions say that: "The construction of the figure is a problem in itself, which can be proposed after the calculation, as an application of Thales' theorem." but I cannot figure how to apply the theorem.
Thanks in advance.
If you call the center of the other circle B, you can argue that $\triangle AMO$ and $\triangle BMO$ are congruent due to same lengths on all sides.
Then, $$\angle MOA = \angle MOB = 180/2 = 90$$
Therefore, $MO$ is perpendicular to $AO$.