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