The two small circles (in black) are equal in radius, and tangential to the large circle. They also touch each other at the center of the large circle.
Now, I want to construct a circle (in orange) which is tangential to the inner two, and the larger circle.
How can I construct it?
I tried it by my own. Hoping there are no mistakes.
Draw a line from the center of the orange circle to the center of the lower small circle. Call the angle from this line the horizontal $\alpha$.
Call the radius of the smaller circle $r$ and the radius of the biggest circle $1$. The radius of the smaller black circles is then obviously $1/2$.
Then you get two equations (from horizontal and vertical distances): $$1=r+(r+1/2)\cos\alpha$$ $$(r+1/2)\sin\alpha=1/2$$ Solving them you get $r=1/3$.
This can be easly constructed:
Draw a horizontal line through the blue center of the big circle and divide it by three. You know how? Then you have the Center of your circle.
[Edit] I used the identity $\cos^2\alpha+\sin^2\alpha=1$. You can skip the step with the angle $\alpha$ and use directly the pythagoras. Which leads to the same.