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 particular
polygon.
More generally, the locus of the centroid of any number of the 
points is a circle (Casey 1888).
See also
Bicentric Polygon, Poncelet's Porism, Weill Point
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References
Casey, J. Quart. J. Pure Appl. Math. 5, 44, 1862.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 164, 1888.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.
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Cite this as:
Weisstein, Eric W. "Weill's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeillsTheorem.html