1. ## IMO Problem II

This problem comes from this year's International Mathematical Olympiad , comparing with another geometry problem on the next day (problem 4) , this seems a little harder . Enjoy !

Let $\displaystyle I$ be the incentre of triangle $\displaystyle ABC$ and let $\displaystyle \Gamma$ be its circumcircle. Let the line $\displaystyle AI$ intersect $\displaystyle \Gamma$ again at $\displaystyle D$ . Let $\displaystyle E$ be a point on the arc $\displaystyle BDC$ and $\displaystyle F$ a point on the side $\displaystyle BC$ such that

$\displaystyle \angle BAF = \angle CAE < \frac{1}{2} \angle BAC$ .

Finally , let $\displaystyle G$ be the midpoint of the segment $\displaystyle IP$ . Prove that the lines $\displaystyle DG$ and $\displaystyle EI$ intersects on $\displaystyle \Gamma$ .

2. sir, sorry for the late,late reply. now only i saw this thread

Originally Posted by simplependulum
Finally , let $\displaystyle G$ be the midpoint of the segment $\displaystyle IP$ .
could you tell me what is $\displaystyle P$ here?

could you tell me what is $\displaystyle P$ here?
4. Yes , it is a typo . It should be $\displaystyle IF$ not $\displaystyle IP$ .