Mathematical and Physical Journal
for High Schools
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# Problem B. 4730. (September 2015)

B. 4730. The circles $\displaystyle k_1$ and $\displaystyle k_2$ touch at point $\displaystyle E$. Points $\displaystyle X_i$ and $\displaystyle Y_i$ are marked on each circle $\displaystyle k_i$ ($\displaystyle i = 1,2$) such that the two lines $\displaystyle X_iY_i$ intersect each other on the common interior tangent of the circles. Prove that the line connecting the centres of circles $\displaystyle X_1X_2E$ and $\displaystyle Y_1Y_2E$, and the other line connecting the centres of circles $\displaystyle X_1Y_2E$ and $\displaystyle X_2Y_1E$ also intersect each other on the common interior tangent of the circles.

Proposed by K. Williams, Szeged

(5 pont)

Deadline expired on October 12, 2015.

### Statistics:

 17 students sent a solution. 5 points: Baran Zsuzsanna, Bodolai Előd, Cseh Kristóf, Döbröntei Dávid Bence, Imolay András, Kerekes Anna, Lajkó Kálmán, Polgár Márton, Schrettner Bálint, Varga-Umbrich Eszter. 4 points: Barabás Ábel, Bukva Balázs, Gáspár Attila. 3 points: 2 students. 0 point: 2 students.

Problems in Mathematics of KöMaL, September 2015