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## Inscribed and Circumscribed Figures

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

# Level 5

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.

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 $$ADBC$$ are trapeziums.

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

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 $$S_{1}$$.

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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