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 $ .