Problem

Source: JMO 2019 Problem 4, by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal

Tags: geometry, USAJMO, 2019 USAJMO, excircle, altitude, Olympiad, Olympiad Geometry



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