Weill's theorem states that, given the incircle and circumcircle of a bicentric
polygon of 
sides, the centroid of the tangent points on the incircle
is a fixed point 
,
known as the Weill point, independent of the orientation of the polygon.
For a triangle 
, the Weill point 
is the triangle centroid
of the contact triangle 
. The Weill point is Kimberling
center 
,
and has equivalent triangle center functions
 |  |  |
(1)
|
 |  |  |
(2)
|
If 
, 
and 
are the circumcenter, incenter,
and Weill point of a triangle 
, then 
lies on the line 
and
 |
(3)
|
where 
and 
are the inradius
and circumradius of 
.
