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 any with incenter and -excenter , let be the midpoint of on 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.
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 However, Hence, is isosceles, so . The rest of the proof proceeds along these lines.