Angle bisector
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Contents
[hide]Angle Bisector
For an angle , the (internal) angle bisector of
is the line from
such that the angle between this line and
is congruent to the angle between this line and
:

An angle also has an external angle bisector, which bisects external angle
:
The external angle is defined by
and the two angle bisectors are perpendicular to each other.
Features of Angle Bisectors
- The distances from a point on an angle bisector to both of its sides are equal.
- The angle bisectors are the locus of points which are equidistant from the two sides of the angle.
- A reflection about either angle bisector maps the two sides of the angle to each other.
- In a triangle, the Angle Bisector Theorem gives the ratio in which the angle bisector cuts the opposite side.
- In a triangle, the internal angle bisectors (which are cevians) all intersect at the incenter of the triangle. The internal angle bisector of one angle and the external angle bisectors of the other two angles all intersect at an excenter of the triangle.
- A bisector of an angle can be constructed using a compass and straightedge.
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Triangle ![]() |
Proofs
Angle Bisector Locus Theorem
Theorem: A point lies on the internal angle bisector of if and only if it is equidistant from the sides
and
.
Proof:
Let point lie on the angle bisector of
. Drop perpendiculars from
to lines
and
, meeting them at points
and
respectively.
Since
and both triangles
and
are right triangles with a shared hypotenuse
and equal angles, they are congruent by AAS. Hence,
Conversely, if is equidistant from
and
, then
, and
lies on the angle bisector of
.
Angle Bisector Theorem
Theorem: In triangle , if the internal angle bisector of
intersects
at point
, then
Proof:
Let , and draw the angle bisector
.
Construct a line through parallel to
, and let it intersect the extension of
at point
.
Since
(by alternate interior angles), triangles
and
are similar by AA.
Therefore,
Internal and External Angle Bisectors Are Perpendicular
Theorem: The internal and external angle bisectors of any angle are perpendicular.
Proof:
Let the internal and external angle bisectors divide and its supplement into two equal parts.
The internal angle is , and the external angle is
.
So the internal bisector makes an angle of with one side, and the external bisector makes an angle of
with the same side. Adding them:
so the two bisectors are perpendicular.