OMCPLP 2022 Problem 5

Tow circumferences of radius $R_1$ and $R_2$ are tangent externally between each other. Besides that, they are both tangent to a semicircle with radius of 1, as shown in the figure. (Diagram is in the attachment)

a) If $A_1$ and $A_2$ are the tangency points of the two circumferences with the diameter of the semicircle, find the length of $\overline{A_1 A_2}$ .

b) Prove that $R_{1}+R_{2}=2\sqrt{R_{1}R_{2}}(\sqrt{2}-\sqrt{R_{1}R_{2}})$ .

Attachments:

2022-11-03 20.33.pdf