Poncelet

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.

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.

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

References

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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Poncelet's Great Theorem

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References