Mathematical and Physical Journal
for High Schools
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# Problem B. 4670. (December 2014)

B. 4670. Let $\displaystyle A_1$, $\displaystyle B_1$ and $\displaystyle C_1$ be the midpoints of the sides of a triangle $\displaystyle ABC$. Drop a perpendicular from $\displaystyle A_1$ to the angle bisector drawn from vertex $\displaystyle A$, from $\displaystyle B_1$ to the angle bisector drawn from vertex $\displaystyle B$, and from $\displaystyle C_1$ to the angle bisector drawn from vertex $\displaystyle C$. Let $\displaystyle A_2$ denote the intersection of the perpendiculars from $\displaystyle B_1$ and from $\displaystyle C_1$. The points $\displaystyle B_2$ and $\displaystyle C_2$ are obtained in a similar way. Show that the lines $\displaystyle A_1A_2$, $\displaystyle B_1B_2$ and $\displaystyle C_1C_2$ are concurrent.

Suggested by Zs. Sárosdi, Veresegyház

(3 pont)

Deadline expired on January 12, 2015.

### Statistics:

 36 students sent a solution. 3 points: Csépai András, Gál Boglárka, Geng Máté, Gyulai-Nagy Szuzina, Heinc Emília, Juhász 326 Dániel, Kerekes Anna, Khayouti Sára, Nagy Dávid Paszkál, Németh 123 Balázs, Polgár Márton, Vankó Miléna, Várkonyi Dorka, Williams Kada. 2 points: Andó Angelika, Ratkovics Gábor, Sal Kristóf, Szakály Marcell, Szász Dániel Soma. 1 point: 8 students. 0 point: 9 students.

Problems in Mathematics of KöMaL, December 2014