Školjka

Let $ABC$ be a fixed acute-angled triangle. Consdeir some points $E$ and $F$ lying on the sides $AC$ and $AB$, respecitvely, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ ant $S$ and $T$, respectively. We call the pair $(E, F)$ \emph{interesting}, the quadrilateral $KSAT$ is cyclic. Suppose that the pairs $(E_1, F_1)$ and $(E_2, F_2)$ are interesting. Prove that $$ \frac{E_1 E_2}{AB} = \frac{F_1 F_2}{AC} \text{.} $$ \begin{flushright}\emph{(Iran)}\end{flushright}