Problem A. 648. (September 2015)

A. 648. In the acute angled triangle \(\displaystyle ABC\), the midpoints of the sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) are \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\), respectively. The foot of the altitude of the triangle starting from \(\displaystyle C\) is \(\displaystyle T_1\). On some line, passing through point \(\displaystyle C\) but not containing \(\displaystyle T_1\), the feet of the perpendiculars starting from \(\displaystyle A\) and \(\displaystyle B\) are \(\displaystyle T_2\) and \(\displaystyle T_3\), respectively. Prove that the circle \(\displaystyle DEF\) passes through the center of the circle \(\displaystyle T_1T_2T_3\).

Proposed by: Bálint Bíró, Eger

(5 pont)

Deadline expired on October 12, 2015.

Statistics:

15 students sent a solution.
5 points:	Adnan Ali, Baran Zsuzsanna, Bukva Balázs, Cseh Kristóf, Gáspár Attila, Kocsis Júlia, Kovács 162 Viktória, Lajkó Kálmán, Schrettner Bálint, Williams Kada.
4 points:	Bodnár Levente, Kiss Dorina, Kovács 246 Benedek, Szabó 789 Barnabás, Szebellédi Márton.

Problems in Mathematics of KöMaL, September 2015