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"ruler and compass" construction of a pentagon 

With
a compass draw a circle. Choose a point A on the circle. With the same
radius as the circle, put the point of the compass at A and draw an arc
intersecting the circle at B. With the same radius and centre B draw
and arc that intersect the circle at C. Continue to construct points A,
B, C, D, E and F on the circle. Join A, C and E to form an equilateral
triangle. 

Draw the line segment AD
intersecting CE at P. Draw the line segment EB intersecting AC at Q. P
is the midpoint of CE and Q is the midpoint of AC. Join P and Q and
extend this line to meet the circle at R. ..................................................................................................................................................................................................................


We are now going to construct a pentagon with side length QR and diagonal length PQ. Put the compass at R and with radius QR draw an arc. With the compass at Q and radius PQ draw an arc to intersect the first arc at S. Join RS. 

Put the compass at S and with radius QR draw an arc. With the compass at R and radius PQ draw an arc to intersect the first arc at T. Join ST.  
Repeat once more with the compass at T to construct the point U and then QRSTU is a pentagon. 

An amateur mathematician
named George Odom discovered the construction and sent it to the famous
Canadian geometer H.S.M. Coxeter. Coxeter, amazed that it had not been
discovered long ago, submitted it in Odom's name to the American
Mathematical Monthly where it appeared in the mid 1980's. The featured
solution consisted of a picture accompanied by one word: "Behold!" 

Source of this information was obtained from a web site: http://mathcentral.uregina.ca/QQ/database/QQ.09.02/mary1.html 
In case you would want to explore different methods of constructing a pentagon  here is a link to a web site that describes seven different ways to construct a pentagon. http://www.geocities.com/robinhuiscool/Pentagon.html 