2023 USAMO Problems/Problem 6

Problem

Let ABC be a triangle with incenter $I$ and excenters $I_a$ , $I_b$ , $I_c$ opposite $A$ , $B$ , and $C$ , respectively. Given an arbitrary point $D$ on the circumcircle of $\triangle ABC$ that does not lie on any of the lines $IIa$ , $I_bI_c$ , or $BC$ , suppose the circumcircles of $\triangle DIIa$ 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 $\angle BAD = \angle EAC$ .