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

Find the area of the brown colored region (outer region) if the radius of each circle is 4 units. Give your answer to 5 decimal places.

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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 \(ABCD.\)

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An inscribed Hexagon has sides \( AF = FE = ED = 4 \), \( AB = BC = CD = 2 \). Furthermore, \(ADEF\) and \(ADBC\) are trapeziums.

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

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In the figure there are three semicircles of diameters \( AB \), \( AC \), \( BC \) and two circles \( S_{1} \) and \( S_{2} \) which are touching to the line \( CF \) ( \( CF \) is perpendicular to \( AB \) ). Both circles have same radius and are called Archimedean Twins. Now if \( AB = 10 \), \( AC = 6 \) and \( BC = 4 \) then find the radius of circle \( S_{1} \).

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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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