Exterior Angle Bisector -- from Wolfram MathWorld
- ️Weisstein, Eric W.
The exterior angle bisectors (Johnson 1929, p. 149), also called the external angle bisectors (Kimberling 1998, pp. 18-19), of a triangle 
are the lines bisecting the angles
formed by the sides of the triangles and their extensions, as illustrated above.
Note that the exterior angle bisectors therefore bisect the supplementary angles of the interior angles, not the entire exterior angles.
There are therefore three pairs of oppositely oriented exterior angle bisectors. The exterior angle bisectors intersect pairwise in the so-called excenters 
, 
, and 
. These are the centers of the excircles,
i.e., the three circles that are externally tangent to the sides of the triangle
(or their extensions).
The points determined on opposite sides of a triangle 
by an angle bisector
from each vertex lie on a straight line if either (1) all or (2) one out of the three
bisectors is an external angle bisector (Johnson 1929, p. 149; Honsberger 1995).
The trilinear coordinates of the points 
,
,
and 
are given by 
,
,
and 
,
respectively.
See also
Angle Bisector, Excenter, Excircles, Exterior Angle, Isodynamic Points
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References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 12, 1967.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 149-150, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.
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Cite this as:
Weisstein, Eric W. "Exterior Angle Bisector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExteriorAngleBisector.html