Learn about packing shapes neatly into others, and the resulting geometric properties.

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 \(XY\).

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