Circle inscribed in a semicircle ![enter image description here]( There is black semicircle, which radius is $ R $. The red circle is tangentially inward to the semicircle and to the diameter in its center. The yellow one is tangent externally to the red circle, internally to the semicircle and tangent to the diameter of the semicircle. My question is : What is the relationship between $R$ and the radius $r$ of yellow one? My attempts : I tried to use similarity of triangles, but always I had the third unknown number and two equations. I am sure that the radius of the red one is $ 0.5R$

**HINT**

Consider the Figure below and note that

1. $\overline{O_2H} = \frac{R}{2}-r$;
2. $\overline{O_2O_3} = \frac{R}{2} + r$;
3. $\overline{HO_1} = r$;
4. $\overline{O_1O_3} = R-r$;



Use Pythagorean theoerm on $\triangle O_2HO_3$ and on $\triangle O_1HO_3$ to determine $\overline{HO_3}$ in two ways and thus get an equation, which, once solved, will give you the desider result of $r$ as a function of $R$.

![enter image description here](