So this question is actually the Circle covering proof by goodman and goodman. But I am not able to find the research paper and few websites that I found are not accessible. If anyone can explain how it is proved I will be thankful;
Suppose segments {$B_{r_1}{(a_1)}$, . . . , $B_{r_n}{(a_n)}$} form a non-separable family of segments. Prove that the segment $B_{r_1+···+r_n}$ (a), where $a =\frac{r_1a_1+···+r_na_n}{r_1+···+r_n}$, covers these segments
What I thought:I considered the various balls of different radius i.e $B_{r_1}{(a_1)}$, . . . , $B_{r_n}{(a_n)}$ centered at $a_1,...a_n$ respectively and projected them on to a line which I believe will make it easy for solving. Now if the intersection of all segment is non-empty then their projection also will be non-empty. Over here I can apply Helly's theorem and can say that all the segments share a common point. But how will I prove that the common point is equal to the point given in the question?