1) Construct first the polar line a' of point A w.r. to the conic a.

2) Construct then the conjugate points A

3) A* is the intersection point of AA

4) Pascal's theorem asserts the collinearity of A, A

5) a'' is the polar line of A* with respect to the conic (a). By its construction, every line through A* intersects the polygon p in two points C, C' (not drawn) and the line a'' at a point C'', such that (CC'A*C'')=-1, i.e. a harmonic division.

6) B

7) The conic B through the points B

8) Thus, polygon p is simultaneously inscribed in a and circumscribed on b. By its construction, the lines joining opposite vertices of p meet at A*. In general Brianchon's theorem asserts, that the same is true for every polygon circumscribed on a conic.

9) p is an instance of a "Poncelet" polygon i.e. a polygon simultaneously inscribed and circumscribed on two conics. Poncelet's "great" theorem asserts that if for two given conics a, b, such a polygon exists, then there exist infinitely many.

Some additional remarks:

10) A, B

11) There is a (involutive) projectivity F, interchanging points of the pair (A

12) F leaves the conics a, b invariant. It leaves also invariant every family-member of the conic-family (J) generated by a and b.

13) A and a' are respectively pole-polar of every conic of the family (J).

14) A and A* singular points of (J). a'' considered as a double line is a singular member of J.

15) The transformation F maps a Poncelet polygon (s) to a Poncelet polygon (s'). There are exactly two such polygons that remain invariant under F. One is s

16) The construction could start with an involutive F leaving invariant a family of conics (J), pick a member (a) of (J) and consider one of the two F-invariant Poncelet hexagons inscribed in (a).

Brianchon.html

Brianchon2.html

Brianchon3.html

Harmonic.html

MaximalRegInEllipse.html

PappusLines.html

Pascal.html

Pascal2.html

Symmetric_hexagons.html

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