Problem B. 4790. (April 2016)

B. 4790. In a scalene triangle \(\displaystyle ABC\), a Thales circle is drawn over each median. The Thales circles of the medians drawn from vertices \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) intersect the circumscribed circle of the triangle \(\displaystyle ABC\) again at points \(\displaystyle A_1\), \(\displaystyle B_1\), and \(\displaystyle C_1\), respectively. Prove that the perpendiculars drawn to \(\displaystyle AA_1\) at \(\displaystyle A\), to \(\displaystyle BB_1\) at \(\displaystyle B\) and to \(\displaystyle CC_1\) at \(\displaystyle C\) are concurrent.

Proposed by K. Williams, Szeged, Radnóti M. Gimn.

(5 pont)

Deadline expired on May 10, 2016.

Statistics:

30 students sent a solution.
5 points:	Baran Zsuzsanna, Bodolai Előd, Borbényi Márton, Cseh Kristóf, Döbröntei Dávid Bence, Gáspár Attila, Hansel Soma, Horváth András János, Imolay András, Kerekes Anna, Keresztes László, Keresztfalvi Bálint, Klász Viktória, Kosztolányi Kata, Lajkó Kálmán, Matolcsi Dávid, Nagy Dávid Paszkál, Németh 123 Balázs, Németh 417 Tamás, Nguyen Viet Hung, Polgár Márton, Schrettner Bálint, Tóth Viktor, Váli Benedek.
4 points:	Bukva Balázs, Radnai Bálint.
3 points:	2 students.
0 point:	2 students.

Problems in Mathematics of KöMaL, April 2016