Question 90
Let $k$ be the circumcircle of $\triangle ABC$ and $D$ be a point on the arc $\widehat{AB}$, which does not contain $C$. Denote by $I_{A}$ and $I_{B}$ the incenters of $\triangle ADC$ and $\triangle BDC$, respectively. Prove that the circumcircle of $\triangle I_{A} I_{B} C$ is tangent to $k$ if and only if
$$
\frac{AD}{BD} = \frac{AC + CD}{BC + CD}
$$
$$
\frac{AD}{BD} = \frac{AC + CD}{BC + CD}
$$