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
- Let
be the feet of the perpendiculars from the vertices
of triangle
. Let
be the circumcenter
. Prove that
\[ OA \perp FE . \] (<url>viewtopic.php?search_id=1224374835&t=45647 Source</url>)
Intermediate
- In triangle
, let the
-excircle touch
at
. Let the
-excircle of triangle
touch
at
and let the
-excircle of triangle
touch
at
. Is
true for all triangles
? (<url>viewtopic.php?t=167688 Source</url>)
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)
- Let
be a parallelogram. A variable line
passing through the point
intersects the rays
and
at points
and
, respectively. Let
and
be the centres of the excircles of triangles
and
, touching the sides
and
, respectively. Prove that the size of angle
does not depend on the choice of
. (Source)