Problem C. 1254. (December 2014)

C. 1254. In a triangle \(\displaystyle ABC\), \(\displaystyle T\) is the foot of the altitude drawn from \(\displaystyle C\), and \(\displaystyle AT=3BT\). Let \(\displaystyle F\) denote the midpoint of \(\displaystyle AB\), and let \(\displaystyle D\) denote the point of altitude \(\displaystyle CT\) where \(\displaystyle AB\) subtends a right angle. Prove that if the orthocentre of triangle \(\displaystyle ABC\) coincides with the centroid of triangle \(\displaystyle FBD\) then \(\displaystyle AD\) bisects the angle \(\displaystyle BAC\).

(5 pont)

Deadline expired on January 12, 2015.

Statistics:

117 students sent a solution.
5 points:	76 students.
4 points:	17 students.
3 points:	11 students.
2 points:	7 students.
1 point:	5 students.
0 point:	1 student.

Problems in Mathematics of KöMaL, December 2014