Geometry
# Inscribed and Circumscribed Figures

Suppose $ABCD$ is a cyclic quadrilateral with side $AD$ being a diameter of length $d$, sides $AB$ and $BC$ both having length $a$ and side $CD$ having length $b$ such that $a,b,d$ are all positive integers with $a \ne b.$

Determine the minimum possible perimeter of $ABCD.$

An inscribed Hexagon has sides $AF = FE = ED = 4$, $AB = BC = CD = 2$. Furthermore, $ADEF$ and $ADCB$ are trapeziums.

If the length of chord $AD$ is $\frac{ m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.

Let $ABCD$ be a cyclic quadrilateral with $\angle ABD = \angle ACD >90^{\circ}$.

Let $P$ and $Q$ be the points on major arc $AD$ satisfying $PA=PC$ and $QB=QD$.

Furthermore, let $X$ and $Y$ be the feet of the perpendiculars from $P$ and $Q$ to line $AD$.

If $AB=20$, $CD=14$, and $BC=16$, then find the length of $XY$.