Problem 2, 2023

Let \(ABC\) be a triangle, let \(D\) be the foot of the altitude from \(A\) to the line \(BC\), and let \(\omega\) be the circle whose diameter is the segment \(AD\). Denote by \(G\) the second common point of \(\omega\) and the circle circumscribed about the triangle \(ABC\), and let the line \(AG\) meet the line \(BC\) at the point \(H\).

Prove that if the circles circumscribed about the triangles \(ABH\) and \(ACH\) meet \(\omega\) once more, at the points \(I\) and \(J\) respectively, then the intersection point of the lines \(BI\) and \(CJ\) lies on \(\omega\).