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Difference between revisions of "Excircle"

(→‎Intermediate: link)
(new problems)
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==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
 +
\[
 +
OA \perp FE .
 +
\]
 +
(<url>viewtopic.php?search_id=1224374835&t=45647 Source</url>)
 
===Intermediate===
 
===Intermediate===
 
*In triangle <math>ABC</math>, let the <math>A</math>-excircle touch <math>BC</math> at <math>D</math>. Let the <math>B</math>-excircle of triangle <math>ABD</math> touch <math>AD</math> at <math>P</math> and let the <math>C</math>-excircle of triangle <math>ACD</math> touch <math>AD</math> at <math>Q</math>. Is <math>\angle P\cong\angle Q</math> true for all triangles <math>ABC</math>? (<url>viewtopic.php?t=167688 Source</url>)
 
*In triangle <math>ABC</math>, let the <math>A</math>-excircle touch <math>BC</math> at <math>D</math>. Let the <math>B</math>-excircle of triangle <math>ABD</math> touch <math>AD</math> at <math>P</math> and let the <math>C</math>-excircle of triangle <math>ACD</math> touch <math>AD</math> at <math>Q</math>. Is <math>\angle P\cong\angle Q</math> true for all triangles <math>ABC</math>? (<url>viewtopic.php?t=167688 Source</url>)
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*<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]])
 
*<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 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>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]])
  
 
==See also==
 
==See also==

Revision as of 09:28, 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

  • $\triangle ABC$ is a triangle. Take points $D, E, F$ on the perpendicular bisectors of $BC, CA, AB$ respectively. Show that the lines through $A, B, C$ perpendicular to $EF, FD, DE$ respectively are concurrent. (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)

See also

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