Question 157
Let $ABC$ be a triangle with $AB \lt AC \lt BC$, and let $D$ be a point in the interior of segment $BC$. Let $E$ be a point on the circumcircle of triangle $ABC$ such that $A$ and $E$ lie on opposite sides of line $BC$ and $\angle BAD = \angle EAC$. Let $I, I_{B}, I_{C}, J_{B}$, and $J_{C}$ be the incentres of triangles $ABC, ABD, ADC, ABE$, and $AEC$, respectively.
Prove that $I_{B}, I_{C}, J_{B}$, and $J_{C}$ are concyclic if and only if $AI, I_{B}J_{C}$, and $J_{B}I_{C}$ concur.
Prove that $I_{B}, I_{C}, J_{B}$, and $J_{C}$ are concyclic if and only if $AI, I_{B}J_{C}$, and $J_{B}I_{C}$ concur.