MathNet 2024-part-01 — 168 problems (EN–VI) Question 11

Let $ABC$ be an acute triangle with $AB \gt AC$. Let $D$, $E$ and $F$ denote the feet of its altitudes on $BC$, $AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$.
Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$.