Consider three circles: C_1: x^2+y^2= r^2; C_2:(x-1)^2+(y-1)^2= r^2; C_3:(x-2)^2+(y-1)^2= r^2 If a line L: y= m x+ c be a common tangent to C_1, C_2…

Consider three circles : $$\begin{aligned} & \mathrm{C}_1: x^2+y^2=\mathrm{r}^2 \\ & \mathrm{C}_2:(x-1)^2+(y-1)^2=\mathrm{r}^2 \\ & \mathrm{C}_3:(x-2)^2+(y-1)^2=\mathrm{r}^2 \end{aligned} $$ If a line $\displaystyle \mathrm{L}: y=\mathrm{m} x+\mathrm{c}$ be a common tangent to $\displaystyle \mathrm{C}_1, \mathrm{C}_2$ and $\displaystyle \mathrm{C}_3$ such that $\displaystyle \mathrm{C}_1$ and $\displaystyle \mathrm{C}_3$ lie on one side of line L while $\displaystyle \mathrm{C}_2$ lies on other side, then the value of $\displaystyle 20\left(\mathrm{r}^2+\mathrm{c}\right)$ is equal to :