Let $ABCD$ be a convex quadrilateral with $AB=5$, $AD=17$, and $CD=6$. If the angle bisector of $\angle{BAD}$ and $\angle{ADC}$ intersect at the midpoint of $BC$, find the area of $ABCD$.
Source: 2023 Hong Kong TST
Tags: geometry
Let $ABCD$ be a convex quadrilateral with $AB=5$, $AD=17$, and $CD=6$. If the angle bisector of $\angle{BAD}$ and $\angle{ADC}$ intersect at the midpoint of $BC$, find the area of $ABCD$.