**A. 690.** In a convex quadrilateral \(\displaystyle ABCD\), the perpendicular drawn from \(\displaystyle A\) to line \(\displaystyle BC\) meets the lines \(\displaystyle BC\) and \(\displaystyle BD\) at \(\displaystyle P\) and \(\displaystyle U\), respectively. The perpendicular drawn from \(\displaystyle A\) to line \(\displaystyle CD\) meets the lines \(\displaystyle CD\) and \(\displaystyle BD\) at \(\displaystyle Q\) and \(\displaystyle V\), respectively. The midpoints of the segments \(\displaystyle BU\) and \(\displaystyle DV\) are \(\displaystyle S\) and \(\displaystyle R\), respectively. The lines \(\displaystyle PS\) and \(\displaystyle QR\) meet at \(\displaystyle E\). The second intersection point of the circles \(\displaystyle PQE\) and \(\displaystyle RSE\), other than \(\displaystyle E\), is \(\displaystyle M\). The points \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\), \(\displaystyle E\), \(\displaystyle M\), \(\displaystyle P\), \(\displaystyle Q\), \(\displaystyle R\), \(\displaystyle S\), \(\displaystyle U\), \(\displaystyle V\) are distinct. Show that the center of the circle \(\displaystyle BCD\), the center of the circle \(\displaystyle AUV\) and the point \(\displaystyle M\) are collinear.

(5 points)

**Deadline expired on 10 March 2017.**