Difference between revisions of "Excircle"

Revision as of 10:47, 5 November 2007

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

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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.

Related Formulas

If the circle is tangent to side $a$ of the circle, the radius is $\frac{2K}{-a+b+c}$ , where $K$ is the triangle's area, and $a,b,c$ are side lengths.
If the circle is tangent to side $b$ of the circle, the radius is $\frac{2K}{a-b+c}$ , where $K$ is the triangle's area, and $a,b,c$ are side lengths.
If the circle is tangent to side $c$ of the circle, the radius is $\frac{2K}{a+b-c}$ , where $K$ is the triangle's area, and $a,b,c$ are side lengths.

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>viewtopic.php?search_id=1224374835&t=45647 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)

@@ Line 20: / Line 20: @@
 ===Olympiad===
-*<math>\triangle ABC</math> is a triangle. Take points <math>D, E, F</math> on the perpendicular bisectors of <math>BC, CA, AB</math> respectively. Show that the lines through <math>A, B, C</math> perpendicular to <math>EF, FD, DE</math> respectively are concurrent. ([[1997 USAMO Problems/Problem 2|Source]])
+*Let <math>ABC</math> be a triangle and let <math>\omega</math> be its incircle. Denote by <math>D_1</math> and <math>E_1</math> the points where <math>\omega</math> is tangent to sides <math>BC</math> and <math>AC</math>, respectively. Denote by <math>D_2</math> and <math>E_2</math> the points on sides <math>BC</math> and <math>AC</math>, respectively, such that <math>CD_2 = BD_1</math> and <math>CE_2 = AE_1</math>, and denote by <math>P</math> the point of intersection of segments <math>AD_2</math> and <math>BE_2</math>. Circle <math>\omega</math> intersects segment <math>AD_2</math> at two points, the closer of which to the vertex <math>A</math> is denoted by <math>Q</math>. Prove that <math>AQ = D_2P</math>. ([[2001 USAMO Problems/Problem 2|Source]])
 *Let <math>ABC</math> be a triangle with circumcircle <math>\omega.</math> Point <math>D</math> lies on side <math>BC</math> such that <math>\angle BAD = \angle CAD.</math> Let <math>I_{A}</math> denote the excenter of triangle <math>ABC</math> opposite <math>A,</math> and let <math>\omega_{A}</math> denote the circle with <math>AI_{A}</math> as its diameter. Circles <math>\omega</math> and <math>\omega_{A}</math> meet at <math>P</math> other than <math>A.</math> The circumcle of triangle <math>APD</math> meet line <math>BC</math> again at <math>Q\, (</math>other than <math>D).</math> Prove that <math>Q</math> lies on the excircle of triangle <math>ABC</math> opposite <math>A</math>. (Source: Problem 13.2 - MOSP 2007)
 *Let <math>ABCD </math> be a parallelogram.  A variable line <math> \ell </math> passing through the point <math>A </math> intersects the rays <math>BC </math> and <math>DC </math> at points <math>X </math> and <math>Y </math>, respectively.  Let <math>K </math> and <math>L </math> be the centres of the excircles of triangles <math>ABX </math> and <math>ADY </math>, touching the sides <math>BX </math> and <math>DY </math>, respectively.  Prove that the size of angle <math>KCL </math> does not depend on the choice of <math> \ell </math>. ([[2005 IMO Shortlist Problems/G3|Source]])

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