PlanetMath: angle bisector as locus

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

angle bisector as locus

(Definition)

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 theorem.)

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

$\displaystyle a_1x+b_1y+c_1 = 0,\quad a_2x+b_2y+c_2 = 0$

(1)

$\displaystyle \frac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}} = \pm\frac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}},$

(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 $\alpha_2$ mean 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.

"angle bisector as locus" is owned by pahio.

(view preamble)

Log in to rate this entry.
(view current ratings)

Cross-references: opposite inverses, slopes, perpendicular, slope angles, divisions, lines, equation, AAS, angle, sides, points, locus, angle bisector
There is 1 reference to this entry.

This is version 6 of angle bisector as locus, born on 2007-06-01, modified 2007-06-07.
Object id is 9492, canonical name is AngleBisectorAsLocus.
Accessed 875 times total.

Classification:

AMS MSC:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact