USAMO 2023 Problem 6

Let $ABC$ be a triangle with incenter $I$ and excenters $I_a$ , $I_b$ , and $I_c$ opposite $A$ , $B$ , and $C$ , respectively. Let $D$ be an arbitrary point on the circumcircle of $\triangle{ABC}$ that does not lie on any of the lines $II_a$ , $I_bI_c$ , or $BC$ . Suppose the circumcircles of $\triangle{DII_a}$ and $\triangle{DI_bI_c}$ intersect at two distinct points $D$ and $F$ . If $E$ is the intersection of lines $DF$ and $BC$ , prove that $\angle{BAD} = \angle{EAC}$ .

Proposed by Zach Chroman