MathNet 2013 Question 53

In equilateral triangle $ABC$, a circle $\omega$ is drawn such that it is tangent to all three sides of the triangle. A line is drawn from $A$ to point $D$ on segment $BC$ such that $AD$ intersects $\omega$ at points $E$ and $F$. If $EF = 4$ and $AB = 8$, determine $|AE - FD|$.

Supplemental illustration (AI-generated):