A conic inside a hexagon

Question

Can you prove or disprove the following claim:

Construct a hexagon circumscribed around a conic section. Intersection points of its non-principal diagonals lie on a new conic section.

GeoGebra applet that demonstrates this claim can be found here.

Answer 1

This is a consequence of Pascal and Brianchon's Theorems.

The intersections of the non-principal diagonals can also be seen as the intersections of the triangles $\triangle{DBF}$ and $\triangle{CAE}$. By Brianchon's theorem, the principal diagonals $EB,FC,AD$ are concurrent at a point $X$. Thus the two triangles are perspective. A converse of Pascal's theorem says that the points of intersection of two perspective triangles lie on a common conic.

Details and more precise statements can be found in Hatton's Projective Geometry, pg 189. There you can find Pascal's theorem, its converse, Brianchon's theorem and proofs.