Question

Three circles, of radii $r,2r$ and $r$ are drawn with their centres on the diagonal of a square, touching one another and the square as shown in the figure below. What is the ratio of the radius of the smaller circles to the side of the square?

Answer 1

Let $AH=AD=x$

$\angle BAH={45}^o$

$\Rightarrow \frac{r}{x}=\tan {45}^o$

$\Rightarrow x=r$

$\Rightarrow AB=\sqrt{A{H}^2+B{H}^2}=r\sqrt{2}$

$BC=r$

$\Rightarrow AC=$ $r\sqrt{2}-r=r(\sqrt{2}-1)$

Similarly, $EF=r(\sqrt{2}-1)$

Length of diagonal $AE=2r+2r+4r+2r(\sqrt{2}-1)$

$=r(6+2\sqrt{2})$

If length of square $=9$

$\Rightarrow$ length of diagonal $=a\sqrt{2}$

$\Rightarrow a\sqrt{2}=r(6+2\sqrt{2})$

$\Rightarrow \frac{r}{a}=\frac{\sqrt{2}}{6+2\sqrt{2}}$.