Perpendicular Bisector


A perpendicular bisector CD of a line segment AB is a line segment perpendicular to AB and passing through the midpoint M of AB (left figure). The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centered at A and B with radius AB and connecting their two intersections. This line segment crosses AB at the midpoint M of AB (middle figure). If the midpoint M is known, then the perpendicular bisector can be constructed by drawing a small auxiliary circle around M, then drawing an arc from each endpoint that crosses the line AB at the farthest intersection of the circle with the line (i.e., arcs with radii AA^' and BB^' respectively). Connecting the intersections of the arcs then gives the perpendicular bisector CD (right figure). Note that if the classical construction requirement that compasses be collapsible is dropped, then the auxiliary circle can be omitted and the rigid compass can be used to immediately draw the two arcs using any radius larger that half the length of AB.


The perpendicular bisectors of a triangle DeltaA_1A_2A_3 are lines passing through the midpoint M_i of each side which are perpendicular to the given side. A triangle's three perpendicular bisectors meet (Casey 1888, p. 9) at a point O known as the circumcenter (Durell 1928), which is also the center of the triangle's circumcircle.

See also

Cathetus, Circumcenter, Circumcircle, Midpoint, Perpendicular, Perpendicular Bisector Theorem, Perpendicular Foot

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Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 19-20, 1928.

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Perpendicular Bisector

Cite this as:

Weisstein, Eric W. "Perpendicular Bisector." From MathWorld--A Wolfram Web Resource.

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