Problem A. 621. (September 2014)
A. 621. In an acute triangle \(\displaystyle ABC\), the feet of the altitudes are \(\displaystyle A_1\), \(\displaystyle B_1\) and \(\displaystyle C_1\), respectively. The midpoint of the side \(\displaystyle BC\) is \(\displaystyle F\). The circle \(\displaystyle BCB_1C_1\) intersects the line segment \(\displaystyle AA_1\) at point \(\displaystyle D\). Let \(\displaystyle T\) be that point on the line segment \(\displaystyle DF\) for which the line \(\displaystyle BT\) is tangent to the circle \(\displaystyle AB_1C_1\). Let the line segment \(\displaystyle C_1F\) meet the lines \(\displaystyle BD\) and \(\displaystyle BT\) at \(\displaystyle P\) and \(\displaystyle Q\), respectively. Show that the quadrilateral \(\displaystyle DPQT\) has an inscribed circle.
(5 pont)
Deadline expired on 10 October 2014.
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