Bisection
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.
Not to be confused with Dissection.The perpendicular bisector of a line segment also has the property that each of its points is from segment AB's endpoints:
equidistant
Area bisectors and perimeter bisectors[edit]
Triangle[edit]
There is an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides; each of these intersects the other two sides so as to divide them into segments with the proportions .[11] These six lines are concurrent three at a time: in addition to the three medians being concurrent, any one median is concurrent with two of the side-parallel area bisectors.
The envelope of the infinitude of area bisectors is a deltoid (broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set).[11] The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one. [1]
The sides of the deltoid are arcs of hyperbolas that are asymptotic to the extended sides of the triangle.[11] The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals i.e. 0.019860... or less than 2%.
A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at (all pass through) the center of the Spieker circle, which is the incircle of the medial triangle. The cleavers are parallel to the angle bisectors.
A splitter of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. A line through the incenter bisects one of the area or perimeter if and only if it also bisects the other.[12]
Parallelogram[edit]
Any line through the midpoint of a parallelogram bisects the area[11] and the perimeter.
Bisectors of diagonals[edit]
Quadrilateral[edit]
If a line segment connecting the diagonals of a quadrilateral bisects both diagonals, then this line segment (the Newton Line) is itself bisected by the vertex centroid.
Volume bisectors[edit]
A plane that divides two opposite edges of a tetrahedron in a given ratio also divides the volume of the tetrahedron in the same ratio. Thus any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron[13][14]: pp.89–90
This article incorporates material from Angle bisector on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.