Question 106
Let $ABCD$ be a rectangle with $AB \gt BC$ and let $\omega$ be its circumscribed circle. Let $E$ and $F$ be, respectively, the intersections (other than $A$) of the bisector of the angle $\widehat{BAD}$ with the side $CD$ and with the circle $\omega$. The perpendicular to $DF$ through $E$ meets the chord $DF$ at $G$ and the arc $DF$ not containing $C$ at the point $H$. Prove that:
(a) the segments $DF$ and $FB$ have the same length;
(b) the triangles $DEG$ and $DHG$ are congruent;
(c) the segments $HF$ and $FC$ are equal.
(a) the segments $DF$ and $FB$ have the same length;
(b) the triangles $DEG$ and $DHG$ are congruent;
(c) the segments $HF$ and $FC$ are equal.