5.2.4.40 Various problems on the plane

Condition: Let \( I \) be the center of the inscribed circle in the triangle \( A B C \), and \( J \) be the center of the excircle. Prove that the circumcircle intersects \( I J \) at its midpoint. (This point, according to the tradition of the Rusanovsky Lyceum, is denoted \(W\), and the statement is part of the trefoil lemma.)