MathNet 2019-part-02 — 184 problems (EN–VI) Question 158

Let $ABC$ be a triangle that is isosceles at $A$ but not right-angled. Let $D$ be the point of $(BC)$ such that $(AD)$ is perpendicular to $(AB)$, and let $E$ be the orthogonal projection of $D$ onto $(AC)$. Finally, let $H$ be the midpoint of $[BC]$.

Prove that $AHE$ is isosceles at $H$.