angle bisector as locusAngleBisectorAsLocus2007-06-01 08:33:282009-02-10 04:33:10Definitionangle bisector% this is the default PlanetMath preamble. as your knowledge
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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
\begin{align}
a_1x+b_1y+c_1 = 0,\quad a_2x+b_2y+c_2 = 0
\end{align}
is
\begin{align}
\frac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}} = \pm\frac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}},
\end{align}
which may be written in the form
\begin{align}
x\sin\alpha_1-y\cos\alpha_1+h_1 = \pm(x\sin\alpha_2-y\cos\alpha_2+h_2)
\end{align}
after performing the divisions in (2) termwise; the angles $\alpha_1$ and $\alpha_2$ \PMlinkescapetext{mean} then the slope angles of the lines.
\textbf{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.