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.
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