Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem A. 655. (November 2015)

A. 655. Two circles, $\displaystyle k_1$ and $\displaystyle k_2$ meet at points $\displaystyle A$ and $\displaystyle B$. Points $\displaystyle C$ and $\displaystyle D$ lie on $\displaystyle k_1$, while points $\displaystyle E$ and $\displaystyle F$ lie on circle $\displaystyle k_2$ in such a way that $\displaystyle A$, $\displaystyle C$, $\displaystyle E$ are collinear and $\displaystyle B$, $\displaystyle D$, $\displaystyle F$ are collinear, too. Points $\displaystyle G$ and $\displaystyle H$ are other two points on lines $\displaystyle ACE$ and $\displaystyle BDF$, respectively. The line $\displaystyle CH$ meets $\displaystyle FG$ and $\displaystyle k_1$ the second time at $\displaystyle I$ and $\displaystyle J$, respectively. The line $\displaystyle DG$ meets $\displaystyle EH$ and $\displaystyle k_1$ the second time at $\displaystyle K$ and $\displaystyle L$, respectively. Circle $\displaystyle k_2$ meets the lines $\displaystyle EHK$ and $\displaystyle FGI$ the second time at $\displaystyle M$ and $\displaystyle N$, respectively. The points $\displaystyle A,B,C,\ldots,N$ are distinct. Show that $\displaystyle I$, $\displaystyle J$, $\displaystyle K$, $\displaystyle L$, $\displaystyle M$ and $\displaystyle N$ are either concyclic or collinear.

(5 pont)

Deadline expired on December 10, 2015.

### Statistics:

 9 students sent a solution. 5 points: Baran Zsuzsanna, Cseh Kristóf, Gáspár Attila, Lajkó Kálmán, Williams Kada. 4 points: Bukva Balázs. 3 points: 2 students. 2 points: 1 student.

Problems in Mathematics of KöMaL, November 2015