Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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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 October 10, 2014.


7 students sent a solution.
5 points:Fehér Zsombor, Saranesh Prembabu, Shapi Topor, Szabó 789 Barnabás, Williams Kada.
1 point:1 student.
0 point:1 student.

Problems in Mathematics of KöMaL, September 2014