Question 63
Let $ABC$ be an equilateral triangle with side length $1$. Points $D$, $E$, $F$ lie inside triangle $ABC$ such that $A$, $E$, $F$ are collinear, $B$, $F$, $D$ are collinear, $C$, $D$, $E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC}$, $Y$ on side $\overline{AB}$, and $Z$ on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ}$, $E$ lies on side $\overline{YZ}$, and $F$ lies on side $\overline{XY}$. Compute $AZ$.