Question 104
Let $ABC$ be an equilateral triangle of side length $15$. Let $A_b$ and $B_a$ be points on side $AB$, $A_c$ and $C_a$ be points on side $AC$, and $B_c$ and $C_b$ be points on side $BC$ such that $\triangle AA_bA_c$, $\triangle BB_cB_a$, and $\triangle CC_aC_b$ are equilateral triangles with side lengths $3$, $4$, and $5$, respectively. Compute the radius of the circle tangent to segments $\overline{A_bA_c}$, $\overline{B_aB_c}$, and $\overline{C_aC_b}$.