Question 16
(1) The triangle $ABC$ is inscribed in a circle. $D$ is the midpoint of the arc $BC$ (not containing $A$), similarly $E$ and $F$. Show that the hexagon formed by the intersection of $ABC$ and $DEF$ has its main diagonals parallel to the sides of $ABC$ and intersecting in a single point.
(2) $EF$ meets $AB$ at $X$ and $AC$ at $Y$. Prove that $AXIY$ is a rhombus, where $I$ is the center of the circle inscribed in $ABC$.
(2) $EF$ meets $AB$ at $X$ and $AC$ at $Y$. Prove that $AXIY$ is a rhombus, where $I$ is the center of the circle inscribed in $ABC$.