The following property ([ChaslesConiques, p. 72], [PonceletApplications, pp. 308-372], [PonceletTraite, pp. 1-66]) is a generalization of the Maclaurin

Braikenridge theorem.

All N sides X

Then its free vertex X

If the points {B

The clue here is a combination of the simple fact noticed in RectHypeRelation.html and the Chasles-Steiner theorem ( Chasles_Steiner.html ). To see it draw two

fixed but arbitrary lines {a

following relations are created:

B

B

....

B

Thus, the correspondence created by composing all these line-perspectivities is a line-projectivity from line a

p = p

Thus, considering the pencils of lines {B*

B

is a projective one, hence, by Chasles-Steiner, the intersection-points X

The figure below shows an example of such a conic for the case of a variable pentagon.

The second claim, on the collinearity, follows immediately since in this case a point of the resulting conic is on the line B

Hence the conic contains the line B

Polygon_Inscribed_Polygon.html .

RectHypeRelation.html

Chasles_Steiner.html

Polygon_Inscribed_Polygon.html

[PonceletApplications] Jean Victor Poncelet

[PonceletTraite] Jean Victor Poncelet

Produced with EucliDraw© |