jmc

Circles ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on ${\omega }_1$, ${\omega }_2$, and ${\omega }_3$ respectively such that $P_1{P}_2=P_2{P}_3=P_3{P}_1$ and line $P_i{P}_{i+1}$ is tangent to ${\omega }_i$ for each $i=1,2,3$, where $P_4=P_1$. See the figure below. The area of $△P_1{P}_2{P}_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$? $\text{(A) }546\text{(B) }548\text{(C) }550\text{(D) }552\text{(E) }554$