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The line bisecting a given Line Segment 
can be constructed geometrically, as illustrated above.
References
Courant, R. and Robbins, H. ``How to Bisect a Segment and Find the Center of a Circle with the Compass Alone.'' §3.4.4 in
What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
Oxford, England: Oxford University Press, pp. 145-146, 1996.
Dixon, R. Mathographics. New York: Dover, p. 22, 1991.