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A Very Funky Equilateral Triangle

Source: 2018 AMC 12B #25

February 16, 2018
AMCAMC 122018 AMC 12B2018 AMCgeometrycircles

Problem Statement

Circles ω1\omega_1, ω2\omega_2, and ω3\omega_3 each have radius 44 and are placed in the plane so that each circle is externally tangent to the other two. Points P1P_1, P2P_2, and P3P_3 lie on ω1\omega_1, ω2\omega_2, and ω3\omega_3 respectively such that P1P2=P2P3=P3P1P_1P_2=P_2P_3=P_3P_1 and line PiPi+1P_iP_{i+1} is tangent to ωi\omega_i for each i=1,2,3i=1,2,3, where P4=P1P_4 = P_1. See the figure below. The area of △P1P2P3\triangle P_1P_2P_3 can be written in the form a+b\sqrt{a}+\sqrt{b} for positive integers aa and bb. What is a+ba+b?
[asy] unitsize(12); pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A; real theta = 41.5; pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1; filldraw(P1--P2--P3--cycle, gray(0.9)); draw(Circle(A, 4)); draw(Circle(B, 4)); draw(Circle(C, 4)); dot(P1); dot(P2); dot(P3); defaultpen(fontsize(10pt)); label("P1P_1", P1, E*1.5); label("P2P_2", P2, SW*1.5); label("P3P_3", P3, N); label("ω1\omega_1", A, W*17); label("ω2\omega_2", B, E*17); label("ω3\omega_3", C, W*17); [/asy]
<spanclass=′latex−bold′>(A)</span>546<spanclass=′latex−bold′>(B)</span>548<spanclass=′latex−bold′>(C)</span>550<spanclass=′latex−bold′>(D)</span>552<spanclass=′latex−bold′>(E)</span>554<span class='latex-bold'>(A) </span>546\qquad<span class='latex-bold'>(B) </span>548\qquad<span class='latex-bold'>(C) </span>550\qquad<span class='latex-bold'>(D) </span>552\qquad<span class='latex-bold'>(E) </span>554
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