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Difference between revisions of "Angle bisector"
 
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{{WotWAnnounce|week=June 6-12}}
 
{{WotWAnnounce|week=June 6-12}}
  
−
For an [[angle]] <math>\angle ABC</math>, the angle bisector of <math>\angle ABC</math> is the line from B such that the angle between this line and <math>BC</math> is equal to the angle between this line and <math>AB</math>.
+
For an [[angle]] <math>\angle BAC</math>, the (internal) angle bisector of <math>\angle BAC</math> is the line from <math>A</math> such that the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>:
  
 
<center>[[Image:Anglebisector.png]]</center>
 
<center>[[Image:Anglebisector.png]]</center>
 +
 +
An [[angle]] <math>\angle BAC</math> also has an external angle bisector, which bisects external angle <math>BAC</math>:
 +
 +
<asy>
 +
import markers;
 +
pair A,B,C,D,E,F;
 +
B=(0,0);
 +
C=(5,0);
 +
A=(4,2);
 +
D=(3,4);
 +
E=(10/3,0);
 +
F=rotate(-90,A)*(E);
 +
draw(A--B--C--cycle,blue);
 +
draw(C--D,blue);
 +
pen p=blue+0.15mm;
 +
draw(A--F,p);
 +
dot(A^^B^^C,red);
 +
label("$A$",A,NE);
 +
label("$B$",B,W);
 +
label("$C$",C,E);
 +
markangle(n=1,radius=20,D,A,F,green);
 +
markangle(n=1,radius=22,F,A,B,green);
 +
</asy>
 +
 +
The external angle is defined by <math>\angle A + \text{ external }\angle A = 180^\circ</math>, and the two angle bisectors are perpendicular to each other.
  
 
== Features of Angle Bisectors ==
 
== Features of Angle Bisectors ==
  
−
[[Image:Incenter.PNG|left|thumb|300px|Triangle ''ABC'' with [[incenter]] ''I'', with [[angle bisector]]s (red), [[incircle]] (blue), and [[inradius|inradii]] (green)]]
+
* 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 [[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.
 +
 
 +
{{asy image|<asy>
 +
size(300);
 +
import markers;
 +
pair excenter(pair A, pair B, pair C){
 +
pair X, Z;
 +
X=A+expi((angle(A-B)+angle(C-A))/2);
 +
Z=C+expi((angle(C-B)+angle(A-C))/2);
 +
return extension(X,A,Z,C);
 +
}
 +
pair X=(0,0), Y=(-10,0), Z=(-3,6);
 +
pair exX=excenter(Z,X,Y), exY=excenter(X,Y,Z), exZ=excenter(Y,Z,X);
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label("X",X,2*(1.5,-1));label("Y",Y,2*(-3,-1));label("Z",Z,(0.6,2));
 +
dot(X^^Y^^Z);
 +
draw(0.3*(X-Y)+X--Y+0.3*(Y-X));draw(0.3*(Y-Z)+Y--Z+0.3*(Z-Y));draw(0.3*(X-Z)+X--Z+0.3*(Z-X));
 +
draw(X+0.3*(X-exX)--exX,orange);draw(Y+0.3*(Y-exY)--exY,orange);draw(Z+0.3*(Z-exZ)--0.6*(exZ-Z)+Z,orange);
 +
draw(Y-0.5*(exX-Y)--Y+0.5*(exX-Y),green);draw(Z-0.5*(exX-Z)--Z+0.5*(exX-Z),green);draw(X-0.5*(exY-X)--X+0.5*(exY-X),green);
 +
pair I=extension(X,exX,Z,exZ);
 +
dot("I",I,(-1,2));
 +
draw(circle(I,length(I-foot(I,X,Y))),blue);
 +
markangle(n=1,3*(Z-X)+X,Z,exX,radius=22,marker(markinterval(stickframe(n=3),true)));
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markangle(n=1,exX,Z,Y,radius=20,marker(markinterval(stickframe(n=3),true)));
 +
markangle(n=1,3*(Y-Z)+Z,Y,exZ,radius=22,marker(markinterval(stickframe(n=2),true)));
 +
markangle(n=1,exZ,Y,X,radius=20,marker(markinterval(stickframe(n=2),true)));
 +
markangle(n=1,3*(X-Y)+Y,X,exY,radius=22,marker(markinterval(stickframe(n=1),true)));
 +
markangle(n=1,exY,X,Z,radius=20,marker(markinterval(stickframe(n=1),true)));
  
−
In a triangle, the angle bisectors (which are [[cevian|cevians]]) will all intersect at the [[incenter]] of the triangle.
+
markangle(n=3,I,Z,X,radius=20);
 +
markangle(n=3,Y,Z,I,radius=22);
 +
markangle(n=2,Z,X,I,radius=20);
 +
markangle(n=2,I,X,Y,radius=22);
 +
markangle(n=1,X,Y,I,radius=22);
 +
markangle(n=1,I,Y,Z,radius=20);
 +
</asy>|center|Triangle <math>\triangle XYZ</math> with [[incenter]] ''I'', [[incircle]] (blue), [[angle bisector]]s (orange),  and [[angle bisector|external angle bisectors]] (green)}}
  
 
==See also==
 
==See also==
−
* [[Angle Bisector Theorem]]
 
 
* [[Cevian]]
 
* [[Cevian]]
 
* [[Geometry]]
 
* [[Geometry]]
 
* [[Stewart's Theorem]]
 
* [[Stewart's Theorem]]
  
−
{{stub}}
 
  
 
[[Category:Geometry]]
 
[[Category:Geometry]]

Latest revision as of 14:27, 19 March 2023

This is an AoPSWiki Word of the Week for June 6-12

For an angle $\angle BAC$, the (internal) angle bisector of $\angle BAC$ is the line from $A$ such that the angle between this line and $\overline{AB}$ is congruent to the angle between this line and $\overline{AC}$:

Anglebisector.png

An angle $\angle BAC$ also has an external angle bisector, which bisects external angle $BAC$:

[asy] import markers; pair A,B,C,D,E,F; B=(0,0); C=(5,0); A=(4,2); D=(3,4); E=(10/3,0); F=rotate(-90,A)*(E); draw(A--B--C--cycle,blue); draw(C--D,blue); pen p=blue+0.15mm; draw(A--F,p); dot(A^^B^^C,red); label("$A$",A,NE); label("$B$",B,W); label("$C$",C,E); markangle(n=1,radius=20,D,A,F,green); markangle(n=1,radius=22,F,A,B,green); [/asy]

The external angle is defined by $\angle A + \text{ external }\angle A = 180^\circ$, 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.

[asy] size(300); import markers; pair excenter(pair A, pair B, pair C){ pair X, Z; X=A+expi((angle(A-B)+angle(C-A))/2); Z=C+expi((angle(C-B)+angle(A-C))/2); return extension(X,A,Z,C); } pair X=(0,0), Y=(-10,0), Z=(-3,6); pair exX=excenter(Z,X,Y), exY=excenter(X,Y,Z), exZ=excenter(Y,Z,X); label("X",X,2*(1.5,-1));label("Y",Y,2*(-3,-1));label("Z",Z,(0.6,2)); dot(X^^Y^^Z); draw(0.3*(X-Y)+X--Y+0.3*(Y-X));draw(0.3*(Y-Z)+Y--Z+0.3*(Z-Y));draw(0.3*(X-Z)+X--Z+0.3*(Z-X)); draw(X+0.3*(X-exX)--exX,orange);draw(Y+0.3*(Y-exY)--exY,orange);draw(Z+0.3*(Z-exZ)--0.6*(exZ-Z)+Z,orange); draw(Y-0.5*(exX-Y)--Y+0.5*(exX-Y),green);draw(Z-0.5*(exX-Z)--Z+0.5*(exX-Z),green);draw(X-0.5*(exY-X)--X+0.5*(exY-X),green); pair I=extension(X,exX,Z,exZ); dot("I",I,(-1,2)); draw(circle(I,length(I-foot(I,X,Y))),blue); markangle(n=1,3*(Z-X)+X,Z,exX,radius=22,marker(markinterval(stickframe(n=3),true))); markangle(n=1,exX,Z,Y,radius=20,marker(markinterval(stickframe(n=3),true))); markangle(n=1,3*(Y-Z)+Z,Y,exZ,radius=22,marker(markinterval(stickframe(n=2),true))); markangle(n=1,exZ,Y,X,radius=20,marker(markinterval(stickframe(n=2),true))); markangle(n=1,3*(X-Y)+Y,X,exY,radius=22,marker(markinterval(stickframe(n=1),true))); markangle(n=1,exY,X,Z,radius=20,marker(markinterval(stickframe(n=1),true))); markangle(n=3,I,Z,X,radius=20); markangle(n=3,Y,Z,I,radius=22); markangle(n=2,Z,X,I,radius=20); markangle(n=2,I,X,Y,radius=22); markangle(n=1,X,Y,I,radius=22); markangle(n=1,I,Y,Z,radius=20); [/asy]

Enlarge.png
Triangle $\triangle XYZ$ with incenter I, incircle (blue), angle bisectors (orange), and external angle bisectors (green)

See also

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