Problem 5, 2009

Let \(S\) be the midpoint of a segment \(AB\), and let \(C\) and \(D\) be points of the semicircle with diameter \(AB\) such that \(C\) lies on the arc \(AD\) and \(\angle CSD = 90^\circ\). Let \(E\) be the point where the lines \(AC\) and \(BD\) meet, and \(F\) the point where the lines \(AD\) and \(BC\) meet. Prove that the vector \(\overrightarrow{EF}\) does not depend on the choice of the points \(C\) and \(D\).