TxO Math Bowl 2024 - Guts Contest - Set 8 Problem 1

Consider a tangential quadrilateral ABCD with AB = 4, BC = 9, CD = 13, and DA = 8. Let I be the center of the inscribed quadrilateral. Draw perpendiculars from I to AB, BC, CD, DA, and let them be H_A, H_B, H_C, H_D respectively. If the length of AH_A to maximize [ABCD]^2 can be written as \frac{m}{n}, where \gcd(m,n) = 1, find m + n.