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## Perpendicular Bisector

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Definition

A perpendicular bisector C D of a line segment A B is a line segment perpendicular to A B and passing through the midpoint M of A B. The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centered at A and B with radius A B and connecting their two intersections. This line segment crosses A B at the midpoint M of A B (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 A B at the farthest intersection of the circle with the line (i.e., arcs with radii A A' and B B' respectively). Connecting the intersections of the arcs then gives the perpendicular bisector C D. 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 A B. The perpendicular bisectors of a triangle Δ A_1 A_2 A_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 at a point O known as the circumcenter, which is also the center of the triangle's circumcircle.

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