Perpendicular bisector construction

The perpendicular bisector or bisector or ( Austrian) bisector is a special line that is being studied in the planar geometry. A generalization to three dimensions is the middle vertical plane.

Definition

The bisector is the set of all points that have the same distance from two given points:

Another definition option is: The bisector is the set of centers of all the circles that pass through two given points.

The bisector is therefore a straight line (i.e., vertical) orthogonal to the line connecting the two points and passing through its center.

Construction

We construct a perpendicular bisector between two given points and, by drawing on these two points with a circle arcs of the same radius, which must be greater than half the distance between the two points. The two points of intersection of these two circles determine a straight line. This line is the perpendicular bisector of the line.

Calculation in the coordinate system

In a two-dimensional Cartesian coordinate system are two points A ( xA / yA ) and B ( xB / yB ) where yA yB with, this is the line equation of the perpendicular bisector:

If yA = yB, this is the (non- functional ) equation:

Perpendicular bisectors of the triangle

The perpendicular bisectors of a triangle intersect in one point, namely the circumcenter of the triangle. This area goes through all the vertices of the triangle (see also: Excellent points in the triangle).

Means the vertical plane

The central vertical plane, and two points in space is the plane that is perpendicular to the connecting line [AB ], and passes through the center of this range, ie, the plane of symmetry of the points.

In analytic geometry we obtain an equation of the vertical plane in normal form by allowing the vector as normal vector and the point M ( with the position vector ) is used as start point: