Excircle

An excircle is a circle tangent to the extensions of two sides of a triangle and the third side.

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

Triangle $\triangle XYZ$ and its excircles.

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 $A$ -excenter lies on the angle bisector of $\angle A$ .

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 $a$ of the triangle, the radius is $\frac{K}{s-a}$ , where $K$ is the triangle's area, and $s = \frac{a+b+c}{2}$ is the semiperimeter.

Problems

Introductory

Let $E,F$ be the feet of the perpendiculars from the vertices $B,C$ of triangle $\triangle ABC$ . Let $O$ be the circumcenter $\triangle ABC$ . Prove that $OA \perp FE .$

(<url>https://artofproblemsolving.com/community/c4h45647 Source</url>)

Intermediate

In triangle $ABC$ , let the $A$ -excircle touch $BC$ at $D$ . Let the $B$ -excircle of triangle $ABD$ touch $AD$ at $P$ and let the $C$ -excircle of triangle $ACD$ touch $AD$ at $Q$ . Is $\angle P\cong\angle Q$ true for all triangles $ABC$ ? (<url>viewtopic.php?t=167688 Source</url>)

Olympiad

Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$ , respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$ , respectively, such that $CD_2 = BD_1$ and $CE_2 = AE_1$ , and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$ . Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$ . Prove that $AQ = D_2P$ . (Source)
Let $ABC$ be a triangle with circumcircle $\omega.$ Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD.$ Let $I_{A}$ denote the excenter of triangle $ABC$ opposite $A,$ and let $\omega_{A}$ denote the circle with $AI_{A}$ as its diameter. Circles $\omega$ and $\omega_{A}$ meet at $P$ other than $A.$ The circumcle of triangle $APD$ meet line $BC$ again at $Q\, ($ other than $D).$ Prove that $Q$ lies on the excircle of triangle $ABC$ opposite $A$ . (Source: Problem 13.2 - MOSP 2007)
Let $ABCD$ be a parallelogram. A variable line $\ell$ passing through the point $A$ intersects the rays $BC$ and $DC$ at points $X$ and $Y$ , respectively. Let $K$ and $L$ be the centres of the excircles of triangles $ABX$ and $ADY$ , touching the sides $BX$ and $DY$ , respectively. Prove that the size of angle $KCL$ does not depend on the choice of $\ell$ . (Source)

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