How to cover a disk with radius $1.01$ with three unit disks?

Question

I need to show that one can cover the disk of radius $1.01$ with three distinct unit disks. But after trying around a bit I am not sure that this is even possible. Could you please give me a hint how to do this?

Answer 1

If you offset the unit disk centers by $d$ in symmetric directions, the circles intersect in pairs at a distance $r$ of the origin such that

$$\left(r-\frac d2\right)^2+\frac{3d^2}4=1.$$

The relevant root is

$$r=\frac{\sqrt{4-3d^2}+d}2$$ and it achieves a maximum when $$d=\frac1{\sqrt3},$$ corresponding to

$$r=\frac2{\sqrt3}>1.01$$

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The minimum decentering is obtained with

$$\left(1.01-\frac d2\right)^2+\frac{3d^2}4=1,$$

or $$d\approx0.02031$$