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angle bisector as locus∗
pahio†
2013-03-21 23:04:16
If 0 < α < 180o , then the angle bisector of α 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
a1 x+b1 y+c1 = 0,
a2 x+b2 y+c2 = 0
(1)
is
a2 x+b2 y+c2
a1 x+b1 y+c1
p
= ± p 2 2 ,
2
2
a1 +b1
a2 +b2
(2)
which may be written in the form
x sin α1 − y cos α1 + h1 = ±(x sin α2 − y cos α2 + h2 )
(3)
after performing the divisions in (2) termwise; the angles α1 and α2 mean then
the slope angles of the lines.
sin α1 ± sin α2
Note. The two lines in (2) are perpendicular, since their slopes
cos α1 ± cos α2
are opposite inverses of each other.
∗ hAngleBisectorAsLocusi
created: h2013-03-21i by: hpahioi version: h39492i Privacy
setting: h1i hDefinitioni h51N20i
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