RO-NMO 2010 Problem 91

In a triangle $ABC$ denote by $D,E,F$ the points where the angle bisectors of $\angle CAB,\angle ABC,\angle BCA$ respectively meet it's circumcircle. a) Prove that the orthocenter of triangle $DEF$ coincides with the incentre of triangle $ABC$ . b) Prove that if $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0$ , then the triangle $ABC$ is equilateral.

Marin Ionescu