USAJMO 2019 Problem 4

Let $ABC$ be a triangle with $\angle ABC$ obtuse. The $A$ -excircle is a circle in the exterior of $\triangle ABC$ that is tangent to side $BC$ of the triangle and tangent to the extensions of the other two sides. Let $E$ , $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$ , respectively. Can line $EF$ be tangent to the $A$ -excircle?

Proposed by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal