Question 27
In a triangle $ABC$ denote by $D$, $E$, respectively $F$, the points where the angle bisectors of $\angle CAB$, $\angle ABC$, respectively $\angle BCA$, meet its circumcircle.
a) Prove that the ortocenter of triangle $DEF$ coincides with the incenter of triangle $ABC$.
b) Prove that if $\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = \overrightarrow{0}$, then the triangle $ABC$ is equilateral.
a) Prove that the ortocenter of triangle $DEF$ coincides with the incenter of triangle $ABC$.
b) Prove that if $\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = \overrightarrow{0}$, then the triangle $ABC$ is equilateral.