Editing Bisection (section)

===Triangle===

====Medians====

Each of the three [[Median (geometry)|medians]] of a triangle is a line segment going through one [[Vertex (geometry)#Of a polytope|vertex]] and the midpoint of the opposite side, so it bisects that side (though not in general perpendicularly). The three medians intersect each other at a point which is called the [[Centroid#Of triangle and tetrahedron|centroid]] of the triangle, which is its [[center of mass]] if it has uniform density; thus any line through a triangle's centroid and one of its vertices bisects the opposite side. The centroid is twice as close to the midpoint of any one side as it is to the opposite vertex.

====Perpendicular bisectors====
{{main|Circumcircle}}
The interior [[perpendicular]] bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the [[circumcenter]] (the center of the circle through the three vertices). Thus any line through a triangle's circumcenter and perpendicular to a side bisects that side.

In an [[acute triangle]] the circumcenter divides the interior perpendicular bisectors of the two shortest sides in equal proportions. In an [[obtuse triangle]] the two shortest sides' perpendicular bisectors (extended beyond their opposite triangle sides to the circumcenter) are divided by their respective intersecting triangle sides in equal proportions.<ref name=Mitchell>Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", ''Forum Geometricorum'' 13, 53-59. http://forumgeom.fau.edu/FG2013volume13/FG201307.pdf</ref>{{rp|Corollaries 5 and 6}}

For any triangle the interior perpendicular bisectors are given by <math>p_a=\tfrac{2aT}{a^2+b^2-c^2},</math> <math>p_b=\tfrac{2bT}{a^2+b^2-c^2},</math> and <math>p_c=\tfrac{2cT}{a^2-b^2+c^2},</math> where the sides are <math>a \ge b \ge c</math> and the area is <math>T.</math><ref name=Mitchell/>{{rp|Thm 2}}