The circle which passes through the points of contact of circles $C_1,C_2 \& C_2,C_3 \& C_3,C_1$ is the incircle of $\mathrm{\Delta }{O}_1{O}_2{O}_3$.

Its centre lies on the angle bisectors of $\mathrm{\Delta }\mathrm{}\mathrm{}{O}_1{O}_2{O}_3$.
Hence, required point is incentre of $\mathrm{\Delta }\mathrm{}\mathrm{}{O}_1{O}_2{O}_3$.
Now, $O_1{O}_2=56, O_2{O}_3=25 \& O_3{O}_1=39$
$\Rightarrow \mathrm{{\rm I}}=\left(\frac{56\times 0+25\times \left(-36\right)+39\times 20}{56+25+39},\frac{25\times 7+56\times \left(-8\right)+39\left(7\right)}{25+56+39}\right)$
$=\left(-\mathrm{1,0}\right)$