Question 109
Let $ABCD$ be a cyclic quadrilateral, and let $E$ and $F$ denote the midpoints of diagonals $[AC]$ and $[BD]$, respectively. If $\{G\} = AB \cap CD$, $\{H\} = AD \cap BC$, prove that:
a) the intersection points of the angle bisectors of $\angle AHB$ and $\angle AGD$ with the sides of the quadrilateral $ABCD$ are the vertices of a rhombus;
b) the center of this rhombus lies on the line $EF$.
a) the intersection points of the angle bisectors of $\angle AHB$ and $\angle AGD$ with the sides of the quadrilateral $ABCD$ are the vertices of a rhombus;
b) the center of this rhombus lies on the line $EF$.