Difference between revisions of "Excircle"
m (wording) |
m (problem) |
||
| Line 4: | Line 4: | ||
== Properties == | == Properties == | ||
| − | For any triangle, there are three unique excircles. | + | 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== | ==Related Formulas== | ||
| − | + | *If the circle is tangent to side <math>a</math> of the circle, the radius is <math>\frac{2K}{-a+b+c}</math>, where <math>K</math> is the triangle's area, and <math>a,b,c</math> are side lengths. | |
| + | *If the circle is tangent to side <math>b</math> of the circle, the radius is <math>\frac{2K}{a-b+c}</math>, where <math>K</math> is the triangle's area, and <math>a,b,c</math> are side lengths. | ||
| + | *If the circle is tangent to side <math>c</math> of the circle, the radius is <math>\frac{2K}{a+b-c}</math>, where <math>K</math> is the triangle's area, and <math>a,b,c</math> are side lengths. | ||
==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
===Intermediate=== | ===Intermediate=== | ||
===Olympiad=== | ===Olympiad=== | ||
| − | *[[1997 USAMO Problems/Problem 2]] | + | *<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. ([[Source|1997 USAMO Problems/Problem 2]]) |
==See also== | ==See also== | ||
Revision as of 18:49, 2 November 2007
An excircle is a circle tangent to the extensions of two sides of a triangle and the third side.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
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.
Related Formulas
- If the circle is tangent to side
of the circle, the radius is 
, where 
is the triangle's area, and 
are side lengths. - If the circle is tangent to side
of the circle, the radius is 
, where is the triangle's area, and are side lengths. - If the circle is tangent to side
of the circle, the radius is 
, where is the triangle's area, and are side lengths.
of the circle, the radius is 
, where 
is the triangle's area, and 
are side lengths.
of the circle, the radius is 
, where 
of the circle, the radius is 
, where Problems
Introductory
Intermediate
Olympiad
is a triangle. Take points 
on the perpendicular bisectors of 
respectively. Show that the lines through 
perpendicular to 
respectively are concurrent. (1997 USAMO Problems/Problem 2)
is a triangle. Take points 
on the perpendicular bisectors of 
respectively. Show that the lines through 
perpendicular to 
respectively are concurrent. (
