# angle bisector as locus

If  $0<\alpha<180^{\mathrm{o}}$,  then the angle bisector of $\alpha$ is the locus of all such points which are equidistant from both sides of the angle (it is proved by using the AAS and SSA theorems).

The equation of the angle bisectors of all four angles formed by two intersecting lines

 $\displaystyle a_{1}x\!+\!b_{1}y\!+\!c_{1}\;=\;0,\qquad a_{2}x\!+\!b_{2}y\!+\!c% _{2}\;=\;0$ (1)

is

 $\displaystyle\frac{a_{1}x\!+\!b_{1}y\!+\!c_{1}}{\sqrt{a_{1}^{2}\!+\!b_{1}^{2}}% }\;=\;\pm\frac{a_{2}x\!+\!b_{2}y\!+\!c_{2}}{\sqrt{a_{2}^{2}\!+\!b_{2}^{2}}},$ (2)

which may be written in the form

 $\displaystyle x\sin\alpha_{1}-y\cos\alpha_{1}+h_{1}\;=\;\pm(x\sin\alpha_{2}-y% \cos\alpha_{2}+h_{2})$ (3)

after performing the divisions in (2) termwise; the angles $\alpha_{1}$ and $\alpha_{2}$ then the slope angles of the lines.

Note.  The two lines in (2) are perpendicular, since their slopes $\displaystyle\frac{\sin\alpha_{1}\pm\sin\alpha_{2}}{\cos\alpha_{1}\pm\cos% \alpha_{2}}$ are opposite inverses of each other.

 Title angle bisector as locus Canonical name AngleBisectorAsLocus Date of creation 2013-03-22 17:10:41 Last modified on 2013-03-22 17:10:41 Owner pahio (2872) Last modified by pahio (2872) Numerical id 11 Author pahio (2872) Entry type Definition Classification msc 51N20 Related topic DistanceFromPointToALine Related topic ConverseOfIsoscelesTriangleTheorem Related topic ConstructionOfTangent Related topic LengthsOfAngleBisectors Related topic Incenter Related topic CenterNormalAndCenterNormalPlaneAsLoci