Question 70
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, 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 $AB$ at $S$ and $T$, respectively. We call the pair $(E, F)$ interesting, if the quadrilateral $KSAT$ is cyclic.
Suppose that the pairs $(E_1, F_1)$ and $(E_2, F_2)$ are interesting. Prove that
$$
\frac{E_1E_2}{AB} = \frac{F_1F_2}{AC}
$$
Suppose that the pairs $(E_1, F_1)$ and $(E_2, F_2)$ are interesting. Prove that
$$
\frac{E_1E_2}{AB} = \frac{F_1F_2}{AC}
$$