The following sections of this BookRags Literature Study Guide is offprint from Gale's For Students Series: Presenting Analysis, Context, and Criticism on Commonly Studied Works: Introduction, Author Biography, Plot Summary, Characters, Themes, Style, Historical Context, Critical Overview, Criticism and Critical Essays, Media Adaptations, Topics for Further Study, Compare & Contrast, What Do I Read Next?, For Further Study, and Sources.
(c)1998-2002; (c)2002 by Gale. Gale is an imprint of The Gale Group, Inc., a division of Thomson Learning, Inc. Gale and Design and Thomson Learning are trademarks used herein under license.
The following sections, if they exist, are offprint from Beacham's Encyclopedia of Popular Fiction: "Social Concerns", "Thematic Overview", "Techniques", "Literary Precedents", "Key Questions", "Related Titles", "Adaptations", "Related Web Sites". (c)1994-2005, by Walton Beacham.
The following sections, if they exist, are offprint from Beacham's Guide to Literature for Young Adults: "About the Author", "Overview", "Setting", "Literary Qualities", "Social Sensitivity", "Topics for Discussion", "Ideas for Reports and Papers". (c)1994-2005, by Walton Beacham.
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The midpoint is defined as a position midway between two extreme points. In mathematics it is the point on a line segment or curvilinear arc that divides it into two parts of equal length. In a right triangle the midpoint of the hypotenuse is equidistant from the three vertices of the triangle.
There are several ways to find the midpoint of a particular figure. Two of the most widely used are the lens method and the Mascheroni construction. The lens method is used to determine the midpoint on a line segment. First a lens using circular arcs is constructed around the line segment so that the line connecting the cusps of the lens is perpendicular to the ends of the line segment. Where the line connecting the cusps of the lens intersects the line segment is the midpoint of that line segment. The Mascheroni construction is more complex but allows one to determine the midpoint of a line segment using only a compass. In about 1797 Mascheroni proved that all constructions possible with a compass and straightedge are possible with a movable compass alone. This construction involves drawing a series of circles starting with the endpoints of the line segment as the centers of the first two circles. Drawing a series of seven circles with varying centers and radii eventually produces the midpoint of the initial line as an intersection between the last two drawn circles. It is a complex construction but allows one to determine the midpoint of the line segment with only a movable compass.
Archimedes developed a theorem that relates the midpoint of an arc on a circle to line segments drawn within the circle. Given the circle below let M be the midpoint of the arc AMB. If point C is chosen at random and point D is chosen such that the line segment MD is perpendicular to AC then the length of AD is equal to the sum of the lengths of DC and BC. AD = DC + BC This is known as Archimedes' midpoint theorem and can serve a variety of uses relating the midpoint of the arc on a circle to specific line segments within.