$${\begin{cases} \frac{R}{f_1} = \frac{|AF'|}{f_2} \implies |AF'| = \frac{R\cdot f_2}{f_1}\\ \frac{R}{f_1} = \frac{|BO'|}{\frac{f_1}{2}} \implies |BO'| = \frac{R}{2}\\ \frac{|AF'|}{|O'D| - f_2} = \frac{r}{\frac{f_1}{2} + |O'D|} \\ \frac{|AF'|}{|O'D| - f_2} = \frac{|BO'|}{|O'D|} \\ \end{cases}}$$