Question 22
We will call a disk without its boundary, i.e., a disk without the points on its circumference, a "hedgehog". The diameter of the hedgehog is the diameter of this circle. We will say that the hedgehog "sits" at the point where the center of the corresponding circle is located. Let us consider a triangle with sides $a, b, c$, and hedgehogs sitting at its vertices. It is known that there exists a point inside the triangle from which one can reach any side of the triangle along a straight trajectory without touching any of the hedgehogs. What is the largest possible sum of the diameters of these hedgehogs?
(Oleksii Masalitin)
(Oleksii Masalitin)