In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through the apex of an angle (that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.

Perpendicular line segment bisector

Definition

• The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly.
• The perpendicular bisector of a line segment ${\displaystyle AB}$ also has the property that each of its points ${\displaystyle X}$ is equidistant from segment AB's endpoints:

(D)${\displaystyle \quad |XA|=|XB|}$.

The proof follows from ${\displaystyle }$ and Pythagoras' theorem:

${\displaystyle |XA|^{2}=|XM|^{2}+|MA|^{2}=|XM|^{2}+|MB|^{2}=|XB|^{2}\;.}$

Property (D) is usually used for the construction of a perpendicular bisector:

Construction by straight edge and compass

In classical geometry, the bisection is a simple compass and straightedge construction, whose possibility depends on the ability to draw arcs of equal radii and different centers:

The segment ${\displaystyle AB}$ is bisected by drawing intersecting circles of equal radius ${\displaystyle r>{\tfrac {1}{2}}|AB|}$, whose centers are the endpoints of the segment. The line determined by the points of intersection of the two circles is the perpendicular bisector of the segment.
Because the construction of the bisector is done without the knowledge of the segment's midpoint ${\displaystyle M}$, the construction is used for determining ${\displaystyle M}$ as the intersection of the bisector and the line segment.

This construction is in fact used when constructing a line perpendicular to a given line ${\displaystyle g}$ at a given point ${\displaystyle P}$: drawing a circle whose center is ${\displaystyle P}$ such that it intersects the line ${\displaystyle g}$ in two points ${\displaystyle A,B}$, and the perpendicular to be constructed is the one bisecting segment ${\displaystyle AB}$.

Equations

If ${\displaystyle {\vec {a}},{\vec {b}}}$ are the position vectors of two points ${\displaystyle A,B}$, then its midpoint is ${\displaystyle M:{\vec {m}}={\tfrac {{\vec {a}}+{\vec {b}}}{2}}}$ and vector ${\displaystyle {\vec {a}}-{\vec {b}}}$ is a normal vector of the perpendicular line segment bisector. Hence its vector equation is ${\displaystyle ({\vec {x}}-{\vec {m}})\cdot ({\vec {a}}-{\vec {b}})=0}$. Inserting ${\displaystyle {\vec {m}}=\cdots }$ and expanding the equation leads to the vector equation

(V) ${\displaystyle \quad {\vec {x}}\cdot ({\vec {a}}-{\vec {b}})={\tfrac {1}{2}}({\vec {a}}^{2}-{\vec {b}}^{2}).}$

With ${\displaystyle A=(a_1,a_2),B=(b_{}}$Zdroj:https://en.wikipedia.org?pojem=Perpendicular_bisector
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