$C,C_1$  and  $C_2$  are circles of radii 5,3, and 2, respectively. $C_1$ and $C_2$  touch each other externally and C internally. If the radius of circle $C_3$  which touches C internally and  $C_1$ and  $C_2$ externally is ‘r’ then $\left[r\right]$ (where $\left[·\right]$ = g.i.f)

 
Let $O,O_1,O_2,$  and $O_3$  be the centers of the circles $C,C_1{C}_2,$  and  $C_3$, respectively, and r  the required radius of  $C_3$.
$O_{1}O=2,O{O}_2=3,O_1{O}_3=r+3,O{O}_3=5-r,O_2{O}_3=r+2$
Applying cosine rule in $\Delta O_{3}O{O}_1,$  we get
$\cos \theta =\frac{{(5-r)}^2+2^2-{(r+3)}^2}{2(5-r)\left(2\right)}$
Applying cosine rule in $\Delta O_{2}O{O}_3,$  we get
$\cos (\pi -\theta )=\frac{{(5-r)}^2+3^2-{(r+2)}^2}{2(5-r)\left(3\right)}$
Eliminating $\theta$ , we get r = 30/19.