Difference between revisions of "Excircle"
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===Olympiad=== | ===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]]) | *<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) | ||
==See also== | ==See also== |
Revision as of 10:13, 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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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.
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. (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)