Problem

Source: Canada Repechage 2022/7 CMOQR

Tags: geometry, Canada, repechage



Let $ABC$ be a triangle with $|AB| < |AC|$, where $| ยท |$ denotes length. Suppose $D, E, F$ are points on side $BC$ such that $D$ is the foot of the perpendicular on $BC$ from $A$, $AE$ is the angle bisector of $\angle BAC$, and $F$ is the midpoint of $BC$. Further suppose that $\angle BAD = \angle DAE = \angle EAF = \angle FAC$. Determine all possible values of $\angle ABC$.