Difference between revisions of "Excircle"
(Fixed Broken Link) 
(mark angles) 

Line 3:  Line 3:  
{{asy image<asy>  {{asy image<asy>  
defaultpen(fontsize(8));  defaultpen(fontsize(8));  
−  pair excenter(pair  +  pair excenter(pair A, pair B, pair C){ 
−  pair  +  pair X, Z; 
−  A  +  X=A+expi((angle(AB)+angle(CA))/2); 
−  C  +  Z=C+expi((angle(CB)+angle(AC))/2); 
−  return extension(A,  +  return extension(X,A,Z,C); 
}  }  
−  pair  +  pair X=(0,0), Y=(10,0), Z=(3,6); 
−  pair  +  pair exX=excenter(Z,X,Y), exY=excenter(X,Y,Z), exZ=excenter(Y,Z,X); 
−  draw(circle(  +  draw(circle(exX,length(exXfoot(exX,Y,Z)))); 
−  draw(circle(  +  draw(circle(exY,length(exYfoot(exY,Z,X)))); 
−  draw(circle(  +  draw(circle(exZ,length(exZfoot(exZ,X,Y)))); 
−  draw((  +  draw((XY)+XY+1.5*(YX));draw((YZ)+YZ+(ZY));draw(2*(XZ)+XZ+2*(ZX)); 
−  label("  +  label("X",X,(1.5,1));label("Y",Y,(2,1));label("Z",Z,N); 
−  dot(  +  label("P",exX,NE); 
−  </asy>rightTriangle <math>\triangle  +  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(exXtangent1,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(exXtangent2,dashed);  
+  
+  markscalefactor=0.1;  
+  draw(rightanglemark(exX,tangent1,Z));  
+  draw(rightanglemark(exX,tangent2,Y));  
+  draw(rightanglemark(exX,(xpart(exX),0),(100,0)));  
+  </asy>rightTriangle <math>\triangle XYZ</math> and its excircles.}}  
+  
== Properties ==  == Properties == 
Revision as of 23:01, 30 June 2020
An excircle is a circle tangent to the extensions of two sides of a triangle and the third side.

Triangle and its excircles. 
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. Any of the three excenters lies on the intersection of two external angle bisectors.
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)