Problem 5, 2019

Let \(ABC\) be a triangle. Points \(D\) and \(E\) lie on the rays \(CA\) and \(CB\) respectively, but not on the sides of the triangle \(ABC\), and are chosen so that

\[ |AD| = |BE| = |AB| . \]

Let \(G\) be the point where the lines \(AE\) and \(BD\) meet, and let \(I\) be the centre of the circle inscribed in the triangle \(ABC\). Prove that the line \(GI\) passes through the midpoint of the side \(AB\).