Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.
Problem
Source: 2021 Cono Sur Shortlist G4 https://artofproblemsolving.com/community/c1088686_cono_sur_shortlist__geometry
Tags: geometry, geometric inequality, excircle