IGO 2020 A3

Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) Note. Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$ Proposed by Morteza Saghafian