IMO Shortlist 2014 problem G6

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Dodao/la: arhiva
7. svibnja 2017.

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) 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{.}