 TOPICS  # Perpendicular Bisector A perpendicular bisector of a line segment is a line segment perpendicular to and passing through the midpoint of (left figure). The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centered at and with radius and connecting their two intersections. This line segment crosses at the midpoint of (middle figure). If the midpoint is known, then the perpendicular bisector can be constructed by drawing a small auxiliary circle around , then drawing an arc from each endpoint that crosses the line at the farthest intersection of the circle with the line (i.e., arcs with radii and respectively). Connecting the intersections of the arcs then gives the perpendicular bisector (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 . The perpendicular bisectors of a triangle are lines passing through the midpoint of each side which are perpendicular to the given side. A triangle's three perpendicular bisectors meet (Casey 1888, p. 9) at a point known as the circumcenter (Durell 1928), which is also the center of the triangle's circumcircle.

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

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## References

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. https://mathworld.wolfram.com/PerpendicularBisector.html