Mathematical and Physical Journal
for High Schools
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# 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