Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$.
Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ (x-a_2)^2+(y-b_2)^2\le r_2\\ (x-a_3)^2+(y-b_3)^2\le r_3\\ (x-a_4)^2+(y-b_4)^2\le r_4\end{cases}$$ then there are $i_1,i_2,i_3$ such that the following (sub-)system of inequalities has no solutions $$ \begin{cases} (x-a_{i_1})^2+(y-b_{i_1})^2\le r_{i_1}\\ (x-a_{i_2})^2+(y-b_{i_2})^2\le r_{i_2}\\ (x-a_{i_3})^2+(y-b_{i_3})^2\le r_{i_3}\end{cases}$$
I know this is can be proved thanks to Helly's theorem ($4$ disks in $\mathbb{R}^2$). Starting from scratch, it's not a difficult task to specialize its proof to this case (Radon's theorem can be stated rather easily in low dimension), but I'd like to know if there exists a more immediate argument to prove this statement.