Find the radius(x) of the circle inscribed in the triangle below

Question

In the figure, calculate $x$, if $a=1$ and $b=9$ (S:$x=3$)

Relationships that I found

Let $d$ = $BD$

$\triangle BCD: 2x+d = BC+CD$

$d^2=BC^2+CD^2$

$\triangle AED:2.1+AD = AE+ED=AE+BC$

$AD^2 = AE^2+BC^2$

Answer 1

Note that all the triangles, including the three containing the three respective circles, are similar, which establishes the ratios $$\frac xb = \frac ax\implies x= \sqrt{ab}=3$$