MathNet 2023-part-06 — 191 problems (EN–VI) Question 168

Let $ABC$ be a triangle and let its incircle, centred at $I$, touch the side $BC$ at $D$. A line through $A$ intersects the lines $BC$, $BI$ and $CI$ at $X$, $Y$ and $Z$, respectively. The circle $(ABC)$ intersects the circles $(AIY)$ and $(AIZ)$ again at $U$ and $V$, respectively. Prove that the points $D$, $U$, $V$ and $X$ are concyclic.

(Fedir Yudin and Mykhailo Shtandenko)


Figure 22