Difference between revisions of "Excircle"
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An '''excircle''' is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. | An '''excircle''' is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. | ||
− | {{image}} | + | |
+ | {{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>|right|Triangle <math>\triangle XYZ</math> and its excircles.}} | ||
+ | |||
+ | |||
== Properties == | == Properties == | ||
− | For any triangle, there are three unique excircles. This follows from the fact that there is one, if any, circle such that three given distinct lines are tangent to it. | + | For any triangle, there are three unique excircles. This follows from the fact that there is one, if any, circle such that three given distinct lines are tangent to it. |
+ | |||
+ | 1) Each excenter lies on the intersection of two [[angle bisector|external angle bisectors]]. | ||
+ | |||
+ | 2) The <math>A</math>-excenter lies on the [[angle bisector]] of <math>\angle A</math>. | ||
+ | |||
+ | == Related Geometrical Objects == | ||
+ | *An exradius is a radius of an excircle of a triangle. | ||
+ | *An excenter is the center of an excircle of a triangle. | ||
==Related Formulas== | ==Related Formulas== | ||
− | If the circle is tangent to side <math>a</math> of the | + | If the circle is tangent to side <math>a</math> of the triangle, the radius is <math>\frac{K}{s-a}</math>, where <math>K</math> is the triangle's area, and <math>s = \frac{a+b+c}{2}</math> is the [[semiperimeter]]. |
==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
*Let <math>E,F</math> be the feet of the perpendiculars from the vertices <math>B,C</math> of triangle <math>\triangle ABC</math>. Let <math>O</math> be the circumcenter <math>\triangle ABC</math>. Prove that <cmath>OA \perp FE .</cmath> | *Let <math>E,F</math> be the feet of the perpendiculars from the vertices <math>B,C</math> of triangle <math>\triangle ABC</math>. Let <math>O</math> be the circumcenter <math>\triangle ABC</math>. Prove that <cmath>OA \perp FE .</cmath> | ||
− | (<url> | + | (<url>https://artofproblemsolving.com/community/c4h45647 Source</url>) |
===Intermediate=== | ===Intermediate=== | ||
Line 21: | Line 66: | ||
==See also== | ==See also== | ||
*[[Incircle]] | *[[Incircle]] | ||
− | + | *[[Circumcircle]] | |
[[Category:Definition]] | [[Category:Definition]] | ||
[[Category:Geometry]] | [[Category:Geometry]] |
Latest revision as of 03:02, 1 July 2020
An excircle is a circle tangent to the extensions of two sides of a triangle and the third side.
|
Triangle ![]() |
Contents
Properties
For any triangle, there are three unique excircles. This follows from the fact that there is one, if any, circle such that three given distinct lines are tangent to it.
1) Each excenter lies on the intersection of two external angle bisectors.
2) The -excenter lies on the angle bisector of
.
Related Geometrical Objects
- An exradius is a radius of an excircle of a triangle.
- An excenter is the center of an excircle of a triangle.
Related Formulas
If the circle is tangent to side of the triangle, the radius is
, where
is the triangle's area, and
is the semiperimeter.
Problems
Introductory
- Let
be the feet of the perpendiculars from the vertices
of triangle
. Let
be the circumcenter
. Prove that
(<url>https://artofproblemsolving.com/community/c4h45647 Source</url>)
Intermediate
- In triangle
, let the
-excircle touch
at
. Let the
-excircle of triangle
touch
at
and let the
-excircle of triangle
touch
at
. Is
true for all triangles
? (<url>viewtopic.php?t=167688 Source</url>)
Olympiad
- Let
be a triangle and let
be its incircle. Denote by
and
the points where
is tangent to sides
and
, respectively. Denote by
and
the points on sides
and
, respectively, such that
and
, and denote by
the point of intersection of segments
and
. Circle
intersects segment
at two points, the closer of which to the vertex
is denoted by
. Prove that
. (Source)
- Let
be a triangle with circumcircle
Point
lies on side
such that
Let
denote the excenter of triangle
opposite
and let
denote the circle with
as its diameter. Circles
and
meet at
other than
The circumcle of triangle
meet line
again at
other than
Prove that
lies on the excircle of triangle
opposite
. (Source: Problem 13.2 - MOSP 2007)
- Let
be a parallelogram. A variable line
passing through the point
intersects the rays
and
at points
and
, respectively. Let
and
be the centres of the excircles of triangles
and
, touching the sides
and
, respectively. Prove that the size of angle
does not depend on the choice of
. (Source)