HMMT 2018 G6

Let $A B C$ be an equilateral triangle of side length 1. For a real number $0<x<0.5$, let $A_{1}$ and $A_{2}$ be the points on side $B C$ such that $A_{1} B=A_{2} C=x$, and let $T_{A}=\triangle A A_{1} A_{2}$. Construct triangles $T_{B}=\triangle B B_{1} B_{2}$ and $T_{C}=\triangle C C_{1} C_{2}$ similarly. There exist positive rational numbers $b, c$ such that the region of points inside all three triangles $T_{A}, T_{B}, T_{C}$ is a hexagon with area

$$\frac{8 x^{2}-b x+c}{(2-x)(x+1)} \cdot \frac{\sqrt{3}}{4}$$

Find $(b, c)$.