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)
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(2)
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If , and are the circumcenter, incenter, and Weill point of a triangle , then lies on the line and
(3)
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where and are the inradius and circumradius of .