Bisectors of Angles between Two Straight Lines/General Form

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in general form as:

\(\ds \LL_1: \ \ \) \(\ds l_1 x + m_1 y + n_1\) \(=\) \(\ds 0\)
\(\ds \LL_2: \ \ \) \(\ds l_2 x + m_2 y + n_2\) \(=\) \(\ds 0\)


The angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$ are given by:

$\dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} } = \pm \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }$


Proof

First we convert $\LL_1$ and $\LL_2$ into normal form:

\(\ds \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) \(=\) \(\ds 0\)
\(\ds \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds x \cos \alpha + y \sin \alpha\) \(=\) \(\ds -\dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) where $\cos \alpha = \dfrac {l_1} {\sqrt { {l_1}^2 + {m_1}^2} }$ and $\sin \alpha = \dfrac {m_1} {\sqrt { {l_1}^2 + {m_1}^2} }$
\(\ds x \cos \beta + y \sin \beta\) \(=\) \(\ds -\dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) where $\cos \beta = \dfrac {l_2} {\sqrt { {l_2}^2 + {m_2}^2} }$ and $\sin \beta = \dfrac {m_2} {\sqrt { {l_2}^2 + {m_2}^2} }$


Then from Bisectors of Angles between Two Straight Lines: Normal Form, the angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$ are given by:

\(\ds x \paren {\cos \alpha - \cos \beta} + y \paren {\sin \alpha - \sin \beta}\) \(=\) \(\ds -\dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } + \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\)
\(\ds x \paren {\cos \alpha + \cos \beta} + y \paren {\sin \alpha + \sin \beta}\) \(=\) \(\ds -\dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } - \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\)
\(\ds \leadsto \ \ \) \(\ds \paren {x \cos \alpha + y \sin \alpha + \dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } } - \paren {x \cos \beta + y \sin \beta + \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} } }\) \(=\) \(\ds 0\)
\(\ds \paren {x \cos \alpha + y \sin \alpha + \dfrac {n_1} {\sqrt { {l_1}^2 + {m_1}^2} } } + \paren {x \cos \beta + y \sin \beta + \dfrac {n_2} {\sqrt { {l_2}^2 + {m_2}^2} } }\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) \(=\) \(\ds \dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\)
\(\ds \dfrac {l_1 x + m_1 y + n_1} {\sqrt { {l_1}^2 + {m_1}^2} }\) \(=\) \(\ds -\dfrac {l_2 x + m_2 y + n_2} {\sqrt { {l_2}^2 + {m_2}^2} }\) substituting back for $\cos \alpha$ and $\sin \alpha$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Bisectors_of_Angles_between_Two_Straight_Lines/General_Form&oldid=513146"