2014 IMO Shortlist G6

Let $A B C$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $A C$ and $A B$, respectively, and let $M$ be the midpoint of $E F$. Let the perpendicular bisector of $E F$ intersect the line $B C$ at $K$, and let the perpendicular bisector of $M K$ intersect the lines $A C$ and $A B$ at $S$ and $T$, respectively. We call the pair $(E, F)$ interesting, if the quadrilateral $K S A T$ is cyclic. Suppose that the pairs $\left(E_{1}, F_{1}\right)$ and $\left(E_{2}, F_{2}\right)$ are interesting. Prove that $$\frac{E_{1} E_{2}}{A B}=\frac{F_{1} F_{2}}{A C} .$$

令$A B C$ 为固定锐角三角形。考虑分别位于$A C$ 和$A B$ 边上的一些点$E$ 和$F$，并令$M$ 为$E F$ 的中点。令$E F$ 的垂直平分线与线$B C$ 相交于$K$，并让$M K$ 的垂直平分线分别与线$A C$ 和$A B$ 相交于$S$ 和$T$。如果四边形 $K SA T$ 是循环的，我们称 $(E, F)$ 为有趣的对。假设 $\left(E_{1}, F_{1}\right)$ 和 $\left(E_{2}, F_{2}\right)$ 是有趣的。证明 $$\frac{E_{1} E_{2}}{A B}=\frac{F_{1} F_{2}}{A C} 。$$