Incenter/excenter lemma
In geometry, the incenter/excenter lemma, sometimes called the Trillium theorem, is a result concerning a relationship between the incenter and excenter of a triangle. Given a triangle with incenter
and
-excenter
, let
be the midpoint of the arc
of the triangle's circumcenter. Then, the theorem states that
is the center of a circle through
,
,
, and
.
The incenter/excenter lemma makes frequent appearances in olympiad geometry. Along with the larger lemma, two smaller results follow: first, ,
,
, and
are collinear, and second,
is the reflection of
across
. Both of these follow easily from the main proof.
Proof
Let ,
,
,
and note that
,
,
are collinear (as
is on the angle bisector).
We are going to show that
, the other cases being similar.
First, notice that
\angle LBI = \angle LBC + \angle CBI
= \angle LAC + \angle CBI
= \angle IAC + \angle CBI
= \half A + \half B.
However,
\angle BIL = \angle BAI + \angle ABI
= \half A + \half B.
Hence, is isosceles.
So
.
The rest of the proof proceeds along these lines.