Let $ABC$ be a triangle and the circle $(I,r)$, be the incircle of the triangle,which is tangent to $BC$ at $D$.Let $w$ be a circle which passes through $B$ and $C$ and that is tangent to $(I,r)$ at $L$.Prove that $A$ is located on the radical axis of the circles $w$ and $(IDL)$
I was thinking about using Protasov's Lemma because it is pretty similar to this problem and the tangency of the circles.
I am pretty sure that $LD$ is the angle bisector of $LD$ but I haven't proved it yet.This is also very similar to Protasov's Lemma.
Comment
May be this idea helps: LE is radical axis and tangent points F, J and I are on a circle, this is possible only when A locates on LE.