Inscribed and Circumscribed Figures

Inscribed and Circumscribed Figures: Level 5 Challenges


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 ABCDABCD is a cyclic quadrilateral with side ADAD being a diameter of length dd, sides ABAB and BCBC both having length aa and side CDCD having length bb such that a,b,da,b,d are all positive integers with ab.a \ne b.

Determine the minimum possible perimeter of ABCD.ABCD.

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

If the length of chord ADAD is mn \frac{ m}{n} , where mm and nn are relatively prime integers, find m+nm + n .

In the figure there are three semicircles of diameters AB AB , AC AC , BC BC and two circles S1 S_{1} and S2 S_{2} which are touching to the line CF CF ( CF CF is perpendicular to AB AB ). Both circles have same radius and are called Archimedean Twins. Now if AB=10 AB = 10 , AC=6 AC = 6 and BC=4 BC = 4 then find the radius of circle S1 S_{1} .

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

Let PP and QQ be the points on major arc ADAD satisfying PA=PCPA=PC and QB=QDQB=QD.

Furthermore, let XX and YY be the feet of the perpendiculars from PP and QQ to line ADAD.

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


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