Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 690. (February 2017)

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 pont)

Deadline expired on March 10, 2017.

### Statistics:

 4 students sent a solution. 5 points: Baran Zsuzsanna, Bukva Balázs, Egri Máté, Williams Kada.

Problems in Mathematics of KöMaL, February 2017