Difference between revisions of "Angle bisector"
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* In a triangle, the [[Angle Bisector Theorem]] gives the ratio in which the angle bisector cuts the opposite side. | * 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 [[cevian|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. | * In a triangle, the internal angle bisectors (which are [[cevian|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 [[Constructions|constructed]] using a compass and straightedge. | ||
| − | [[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with [[incenter]] ''I'', with [[angle bisector]]s (red), [[incircle]] (blue), and [[inradius|inradii]] (green)]] | + | [[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with [[incenter]] ''I'', with [[angle bisector]]s (red), [[incircle]] (blue), and [[inradius|inradii]] (green)]] |
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==See also== | ==See also== | ||
Revision as of 17:22, 27 February 2014
| This is an AoPSWiki Word of the Week for June 6-12 |
For an angle 
, the (internal) angle bisector of is the line from B such that the angle between this line and 
is congruent to the angle between this line and 
.
A given angle also has an external angle bisector, which lies entirely outside the angle. The two angle bisectors are perpendicular to each other.
Features of Angle Bisectors
- 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.
See also
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