Editing Bisection (section)

==Area bisectors and perimeter bisectors==

===Triangle===

There is an infinitude of lines that bisect the [[area]] of a [[triangle]]. Three of them are the [[Median (geometry)|medians]] of the triangle (which connect the sides' midpoints with the opposite vertices), and these are [[Concurrent lines|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 <math>\sqrt{2}+1:1</math>.<ref name=Dunn>{{cite journal | last1 = Dunn | first1 = Jas. A. | last2 = Pretty | first2 = Jas. E. | date = May 1972 | doi = 10.2307/3615256 | issue = 396 | journal = The Mathematical Gazette | jstor = 3615256 | pages = 105–108 | title = Halving a triangle | volume = 56}}</ref> 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 (mathematics)|envelope]] of the infinitude of area bisectors is a [[Deltoid curve|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).<ref name=Dunn/> 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. [http://www.se16.info/js/halfarea.htm]
The sides of the deltoid are arcs of [[hyperbola]]s that are [[asymptotic]] to the extended sides of the triangle.<ref name=Dunn/> The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant for all triangles, and equals <math>\tfrac{3}{4} \log_e(2) - \tfrac{1}{2},</math> i.e. 0.019860... or less than 2%.

A [[Cleaver (geometry)|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 [[Concurrent lines|concur]] at (all pass through) the [[Spieker center|center of the Spieker circle]], which is the [[incircle]] of the [[medial triangle]]. The cleavers are parallel to the angle bisectors.

A [[Splitter (geometry)|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.<ref>Kodokostas, Dimitrios, "Triangle Equalizers," ''[[Mathematics Magazine]]'' 83, April 2010, pp. 141-146.</ref>

===Parallelogram===

Any line through the midpoint of a [[parallelogram]] bisects the area<ref name=Dunn/> and the perimeter.

===Circle and ellipse===

All area bisectors and perimeter bisectors of a circle or other ellipse go through the [[Center (geometry)|center]], and any [[Chord (geometry)|chord]]s through the center bisect the area and perimeter. In the case of a circle they are the [[diameter]]s of the circle.