Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 636. (February 2015)

A. 636. There is given a convex quadrilateral $\displaystyle ABCD$ and a point $\displaystyle P$ in the interior of the triangle $\displaystyle BCD$ in such a way that the quadrilateral $\displaystyle ABPD$ has an inscribed circle, and the three inscribed circles of the quadrilateral $\displaystyle ABPD$, the triangle $\displaystyle BCP$ and the triangle $\displaystyle CDP$, respectively, are pairwise tangent to each other. Denote by $\displaystyle Q$ and $\displaystyle R$ the points tangency on the line segments $\displaystyle BP$ and $\displaystyle DP$, respectively. Let the lines $\displaystyle BP$ and $\displaystyle AR$ meet at $\displaystyle S$, let the lines $\displaystyle DP$ and $\displaystyle AQ$ meet at $\displaystyle T$, and let the lines $\displaystyle BT$ and $\displaystyle DS$ meet at $\displaystyle U$. Show that the line $\displaystyle CU$ bisects the angle $\displaystyle BCD$.

(5 pont)

Deadline expired on March 10, 2015.

### Statistics:

 2 students sent a solution. 5 points: Janzer Barnabás. 1 point: 1 student.

Problems in Mathematics of KöMaL, February 2015