Question 89
Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. Let $A_1, A_2$ be points on side $BC$, $B_1, B_2$ be points on side $CA$, and $C_1, C_2$ be points on side $AB$. Suppose that there exists a point $P$ such that $PA_1A_2$, $PB_1B_2$, and $PC_1C_2$ are congruent equilateral triangles. Find the area of convex hexagon $A_1A_2B_1B_2C_1C_2$.