In a right angled triangle BC = 5, AB= 4, AC = 3. Let S be the circum circle. Let $S_1$ be the circle touching both sides AB and AC and circle S internally. Let $S_2$ be the circle touching the produced sides AB and AC of ${△}^{le}ABC$, and touching the circle S externally If $r_1,r_2$  are radii of circles $S_1$ and $S_2$ respectively then   $r_1{r}_2=$

Let A= (0,0), B= (4,0), c= (0,3)
Circumcenter of circle S is (2,3/2) and circumradius =  $\frac{5}{2}$
Equation of S is ${(x-2)}^2+{(y-3/2)}^2=\frac{25}{4}$ ------------(1)
If circle $S_1$ is having radius $r_1$ and touching AB and AC           
$\Rightarrow$ Its centre is ($r_1,r_2$).
$S_1$ and S touch internally $\Rightarrow \sqrt{{(r_1-2)}^2+{(r_1-3/2)}^2}=|r_1-5/2|$
$\Rightarrow {r_1}^2-2r_1=0\Rightarrow r_1=2$
If $S_2$ is having $r_2$ and touching AB and AC $\Rightarrow$ centre of  $S_2= \left(r_1,r_2\right)$
$S_2$ touches S externally  $\Rightarrow {(r_2-2)}^2+{(r_2-3/2)}^2={(r_2+5/2)}^2$
$\Rightarrow {r_2}^2-12r_2=0\Rightarrow r_2=12\Rightarrow r_1{r}_2=24$