Question 47
Let $ABC$ be an equilateral triangle of side length $1$. For a real number $0 \lt x \lt 0.5$, let $A_1$ and $A_2$ be the points on side $BC$ such that $A_1B = A_2C = x$, and let $T_A = \triangle AA_1A_2$. Construct triangles $T_B = \triangle BB_1B_2$ and $T_C = \triangle CC_1C_2$ 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
$$
\frac{8x^2 - bx + c}{(2-x)(x+1)} \cdot \frac{\sqrt{3}}{4}
$$
Find $(b, c)$.
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
$$
\frac{8x^2 - bx + c}{(2-x)(x+1)} \cdot \frac{\sqrt{3}}{4}
$$
Find $(b, c)$.