Question 12
Let $I$ and $I_A$ be the incenter and $A$-excenter of an acute triangle $ABC$, with $AB\lt AC$. The incircle touches $BC$ at $D$. The line $AD$ meets $BI_A$ and $CI_A$ at $E$ and $F$, respectively. Prove that the circumcircles of triangles $AID$ and $I_AEF$ are tangent.
Notation clarification. The source prints “triangle $IAEF$” and circle $(IAEF)$ although $A,E,F$ are collinear. In both source proofs the circle meant is $(I_AEF)$, through the already defined excenter $I_A$ and the points $E,F$. We explicitly restore that missing subscript here; the original source text is retained in the provenance.
Notation clarification. The source prints “triangle $IAEF$” and circle $(IAEF)$ although $A,E,F$ are collinear. In both source proofs the circle meant is $(I_AEF)$, through the already defined excenter $I_A$ and the points $E,F$. We explicitly restore that missing subscript here; the original source text is retained in the provenance.