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

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 .

See also Weill's Theorem
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References Gallatly, W. "The Weill Point." §238in The
Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 19, 1913. M'Clelland,
W. J. A
Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the
Method of Reciprocation, with Numerous Examples. London: Macmillan, p. 96,
1891. Weill. Liouville's J. (Ser. 3) 4 , 270, 1878. Referenced
on Wolfram|Alpha Weill Point
Cite this as:
Weisstein, Eric W. "Weill Point." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/WeillPoint.html

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