Let $ABC$ be a triangle, $\Gamma$ its circumcircle, $I$ its incenter, and $\omega$ a tangent circle to the line $AI$ at $I$ and to the side $BC$. Prove that the circles $\Gamma$ and $\omega$ are tangent.
Malik Talbi
Let $\omega$ be tangent to the side $BC$ at point $E$ and denote $D$ as the center of arc $BC$ not lying on arc $BC$. By trilium theorem $BD=CD=ID$. Consider inversion centered at $D$ with radius $ID$. Points $B,C,I$ are obviously fixed and inversion swaps line $BC$ with circumcircle $ABC$. Because $\omega$ is tangent to line $DI$ at $I$ this is also a fixed object under inversion. Since it’s tangent to $BC$ it’s also tangent to the circumcircle. QED