[alogo] Poncelet's Great Theorem

This theorem (referred also as "Poncelet's Porism"), considered by many as the culminating point of plane projective geometry, concerns a property of a polygonal line inscribed in a conic (a) and simultaneously tangent to another conic (b). The theorem asserts that if the polygonal line closes for one particular position of the start point A on the conic (a) then it closes for every position of A on that conic! A proof for triangles together with the description of a method that applies to general polygons can be found in Poncelet_Proof.html . A detailed proof of the general case is given by Berger in the reference cited below.

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The theorem specializes for circles in the obvious way. For circles and triangles it reduces to Euler's condition on the radii of two circles: [ R^2 - 2*R*r = d^2 ] . R, r being the radii and d the distance of the centers of the two circles. The theorem asserts that this equation is necessary and sufficient in order that a triangle has the two circles as circumscribed and inscribed respectively. The specialization for circles and [Bicentric] (in EucliDraw called [Bicircular]) quadrangles is also known as [Theorem of Fuss]. The condition is: [ 1/r^2 = 1/(R+d)^2 + 1/(R-d)^2 ]. See the file Bicentric.html for a discussion of properties of these quadrilaterals.

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For an apparently more complicated example of polygons with N sides see the file MaximalRegInEllipse.html .

See Also



Baker, H. F. Plane Geometry New York, Chelsea Publishing Company 1971, p. 276.
Berger, M Geometry II Paris, Springer Verlag, 1987, 16.6 The great Poncelet theorem, p. 203.

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