Question 153
The incircle of the triangle $ABC$ touches the sides $\overline{AB}$ and $\overline{AC}$ in $D$ and $E$, respectively. The excircle of the same triangle opposite to the vertex $A$ touches the rays $AB$ and $AC$ in $F$ and $G$, respectively. Let the bisectors of the internal (external) angles $\angle CBA$ and $\angle ACB$ intersect the line $DE$ ($FG$) at the points $X$ and $Y$ ($Z$ and $W$), respectively.
Prove that the points $X$, $Y$, $Z$ and $W$ lie on the same circle.
Prove that the points $X$, $Y$, $Z$ and $W$ lie on the same circle.