A geometry problem by Tushar Verma

The geometric mean: sqrt(8 x 50} = sqrt(400) = 20 yay!

There are 5 circles here.Let us consider the ratio between the radii is p.So we can write 50=8 p p p p,so p=1.58.So r3=8 P P=20

We can imagine that the circles are homothetic transformations onto each other, so they are all similar with a constant ratio, so the radius of the center circle is the geometric mean of that of the outside circles.

$\{r(k)\},\,k ∈ ℕ ∪ \{0\}\,$ is a G.P.

$\,\frac {r(k+1)}{r(k)}= q,\,$

$q\,$ constant ratio of the G.P.

$\frac {r(k+h)}{r(k)}= q^h;\,\frac {r(2h)}{r(h)}= q^h$ $\frac {r(4)}{r(2)}= q^2;\,\frac {r(2)}{r(0)}= q^2$ $\frac {r(4)}{r(2)}=\frac {r(2)}{r(0)};\,r(2)=\sqrt {r(4)r(0)}$

$r(0)=r_2=50,\,r_1=r(4)=8,\,r_3=r(2),$ $r_3=\sqrt {{r_1}{r_2}}=\sqrt {400}=\boxed {20}$

as Hassan said, the radii are in GP..