Excircle
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. (1997 USAMO Problems/Problem 2)