# angle bisector as locus

If $$, 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

${a}_{1}x+{b}_{1}y+{c}_{1}=\mathrm{\hspace{0.33em}0},{a}_{2}x+{b}_{2}y+{c}_{2}=\mathrm{\hspace{0.33em}0}$ | (1) |

is

$\frac{{a}_{1}x+{b}_{1}y+{c}_{1}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}}}}=\pm {\displaystyle \frac{{a}_{2}x+{b}_{2}y+{c}_{2}}{\sqrt{{a}_{2}^{2}+{b}_{2}^{2}}}},$ | (2) |

which may be written in the form

$x\mathrm{sin}{\alpha}_{1}-y\mathrm{cos}{\alpha}_{1}+{h}_{1}=\pm (x\mathrm{sin}{\alpha}_{2}-y\mathrm{cos}{\alpha}_{2}+{h}_{2})$ | (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
$\frac{\mathrm{sin}{\alpha}_{1}\pm \mathrm{sin}{\alpha}_{2}}{\mathrm{cos}{\alpha}_{1}\pm \mathrm{cos}{\alpha}_{2}}$ 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 |