Question 18
In the plane there are a triangle $\triangle ABC$ and a point $O$; $\Gamma$ is the circumcircle of $\triangle ABC$. Let the line $CO$ meet the line $AB$ at the point $D$, and let the line $BO$ meet the line $CA$ at the point $E$. Let the line $AO$ meet $\Gamma$ again at the point $F$. Let $I$ be the second intersection point of $\Gamma$ and the circumcircle of $\triangle ADE$, let $Y$ be the second intersection point of the line $BE$ with the circumcircle of $\triangle CEI$, and let $Z$ be the second intersection point of the line $CD$ with the circumcircle of $\triangle BDI$. The two tangents to $\Gamma$ at $B$ and $C$ meet at the point $T$. Let the line $TF$ meet $\Gamma$ again at the point $U$, and let $G$ be the reflection of $U$ in the line $BC$.
Prove that the six points $F, I, G, O, Y, Z$ are concyclic.
Prove that the six points $F, I, G, O, Y, Z$ are concyclic.