Editing Bisection (section)

==Angle bisector==
[[File:Bisection construction.gif|right|thumb|Bisection of an angle using a compass and straightedge]]

An '''angle bisector''' divides the [[angle]] into two angles with [[equality (mathematics)|equal]] measures. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle.

The 'interior' or 'internal bisector' of an angle is the line, [[Ray (geometry)|half-line]], or line segment that divides an angle of less than 180° into two equal angles.  The 'exterior' or 'external bisector' is the line that divides the [[supplementary angle]] (of 180° minus the original angle), formed by one side forming the original angle and the extension of the other side, into two equal angles.<ref>[http://mathworld.wolfram.com/ExteriorAngleBisector.html Weisstein, Eric W. "Exterior Angle Bisector." From MathWorld--A Wolfram Web Resource.]</ref>

To bisect an angle with [[straightedge and compass]], one draws a circle whose center is the vertex. The circle meets the angle at two points: one on each leg.  Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector.

The proof of the correctness of this construction is fairly intuitive, relying on the symmetry of the problem. The [[Angle trisection|trisection of an angle]] (dividing it into three equal parts) cannot be achieved with the compass and ruler alone (this was first proved by [[Pierre Wantzel]]).

The internal and external bisectors of an angle are [[perpendicular]]. If the angle is formed by the two lines given algebraically as  <math>l_1x+m_1y+n_1=0</math> and <math>l_2x+m_2y+n_2=0,</math> then the internal and external bisectors are given by the two equations<ref>Spain, Barry. ''Analytical Conics'', Dover Publications, 2007 (orig. 1957).</ref>{{rp|p.15}}

:<math>\frac{l_1x+m_1y+n_1}{\sqrt{l_1^2+m_1^2}} = \pm \frac{l_2x+m_2y+n_2}{\sqrt{l_2^2+m_2^2}}.</math>