Difference between revisions of "Angle bisector"

Revision as of 00:13, 1 July 2020

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

For an angle $\angle ABC$ , the (internal) angle bisector of $\angle ABC$ is the line from B such that the angle between this line and $BC$ is congruent to the angle between this line and $AB$ :

A given angle $\angle ABC$ also has an external angle bisector, which bisects external angle $ABC$ where $\text{internal angle }A + \text{ external angle }A = 180^\circ$ :

$[asy] pair A,B,C,D; B=(0,0); C=(5,0); A=(4,2); D=(3,4); draw(A--B--C--cycle,blue); draw(C--D,blue); [/asy]$

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.

$[asy] defaultpen(fontsize(8)); pair excenter(pair A, pair B, pair C){ pair X, Z; X=A+expi((angle(A-B)+angle(C-A))/2); Z=C+expi((angle(C-B)+angle(A-C))/2); return extension(X,A,Z,C); } pair X=(0,0), Y=(10,0), Z=(3,6); pair exX=excenter(Z,X,Y), exY=excenter(X,Y,Z), exZ=excenter(Y,Z,X); draw(circle(exX,length(exX-foot(exX,Y,Z)))); draw(circle(exY,length(exY-foot(exY,Z,X)))); draw(circle(exZ,length(exZ-foot(exZ,X,Y)))); draw((X-Y)+X--Y+1.5*(Y-X));draw((Y-Z)+Y--Z+(Z-Y));draw(2*(X-Z)+X--Z+2*(Z-X)); label("X",X,(-1.5,-1));label("Y",Y,(2,-1));label("Z",Z,N); label("P",exX,NE); dot(X^^Y^^Z^^exX^^exY^^exZ); draw(exX--(xpart(exX),0),dashed); real slope1=(ypart(Z)-ypart(X))/(xpart(Z)-xpart(X)); pair point1=exX+(1,-1/slope1); pair tangent1=extension(X,Z,point1,exX); draw(exX--tangent1,dashed); real slope2=(ypart(Y)-ypart(Z))/(xpart(Y)-xpart(Z)); pair point2=exX+(1,-1/slope2); pair tangent2=extension(Y,Z,point2,exX); draw(exX--tangent2,dashed); markscalefactor=0.1; draw(rightanglemark(exX,tangent1,Z)); draw(rightanglemark(exX,tangent2,Y)); draw(rightanglemark(exX,(xpart(exX),0),(100,0))); [/asy]$

Triangle $\triangle XYZ$ with incenter I, with angle bisectors (red), incircle (blue), and inradii (green)

[[Image:Incenter.PNG|left|thumb|300px|Triangle ABC with incenter I, with angle bisectors (red), [[external angle bisectorincircle (blue), and inradii (green)]]

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Difference between revisions of "Angle bisector"

Revision as of 00:13, 1 July 2020

Features of Angle Bisectors

See also

@@ Line 1: / Line 1: @@
 {{WotWAnnounce|week=June 6-12}}
-For an [[angle]] <math>\angle ABC</math>, the (internal) angle bisector of <math>\angle ABC</math> is the line from B such that the angle between this line and <math>BC</math> is congruent to the angle between this line and <math>AB</math>.
+For an [[angle]] <math>\angle ABC</math>, the (internal) angle bisector of <math>\angle ABC</math> is the line from B such that the angle between this line and <math>BC</math> is congruent to the angle between this line and <math>AB</math>:
 <center>[[Image:Anglebisector.png]]</center>
-A given angle also has an external angle bisector, which lies entirely outside the angle. The two angle bisectors are perpendicular to each other.
+A given [[angle]] <math>\angle ABC</math> also has an external angle bisector, which bisects external angle <math>ABC</math> where <math>\text{internal angle }A + \text{ external angle }A = 180^\circ</math>:
+{{asy image |<asy>
+pair A,B,C,D;
+B=(0,0);
+C=(5,0);
+A=(4,2);
+D=(3,4);
+draw(A--B--C--cycle,blue);
+draw(C--D,blue);
+</asy>|center|}}
+ The two angle bisectors are perpendicular to each other.
 == Features of Angle Bisectors ==
@@ Line 16: / Line 28: @@
 *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)]]
+{{asy image|<asy>
+defaultpen(fontsize(8));
+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);
+draw(circle(exX,length(exX-foot(exX,Y,Z))));
+draw(circle(exY,length(exY-foot(exY,Z,X))));
+draw(circle(exZ,length(exZ-foot(exZ,X,Y))));
+draw((X-Y)+X--Y+1.5*(Y-X));draw((Y-Z)+Y--Z+(Z-Y));draw(2*(X-Z)+X--Z+2*(Z-X));
+label("X",X,(-1.5,-1));label("Y",Y,(2,-1));label("Z",Z,N);
+label("P",exX,NE);
+dot(X^^Y^^Z^^exX^^exY^^exZ);
+draw(exX--(xpart(exX),0),dashed);
+real slope1=(ypart(Z)-ypart(X))/(xpart(Z)-xpart(X));
+pair point1=exX+(1,-1/slope1);
+pair tangent1=extension(X,Z,point1,exX);
+draw(exX--tangent1,dashed);
+real slope2=(ypart(Y)-ypart(Z))/(xpart(Y)-xpart(Z));
+pair point2=exX+(1,-1/slope2);
+pair tangent2=extension(Y,Z,point2,exX);
+draw(exX--tangent2,dashed);
+markscalefactor=0.1;
+draw(rightanglemark(exX,tangent1,Z));
+draw(rightanglemark(exX,tangent2,Y));
+draw(rightanglemark(exX,(xpart(exX),0),(100,0)));
+</asy>|center|Triangle <math>\triangle XYZ</math> 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), [[external angle bisector[[incircle]] (blue), and [[inradius|inradii]] (green)]]
 ==See also==