Question 104
Let $ABC$ a triangle and $\omega$ its inscribed circle whose points of tangency on $[BC]$, $[CA]$ and $[AB]$ are respectively $D$, $E$ and $F$. We note $\Omega$ the circle tangent to $\omega$ and passing through $B$ and $C$ and $T$ the point of tangency of these two circles. We note $X$ and $Y$ the media of the segments $[DE]$ and $[DF]$. We note $U$ and $V$ the second intersection points of the straight lines $(TX)$ and $(TY)$ with $\Omega$. We also note $Z$ and $W$ the second intersection points of the straight lines $(AB)$ and $(AC)$ with $\Omega$. Show that $(ZU)$, $(WV)$, $(XY)$ and angle bisector $\widehat{BAC}$ are concurrent.