Proposition:To Demonstrate that the opposite vertex lines of a hexagram circumscribed about a conic section pass through a point.
Here is a construction of the Brianchon Point:.

Proof of this theorem is quite a challenge! Dorrie's relies heavily on projection theorems of Steiner and Desargues. For convenience, let a line joining opposite vertices of a hexagram be called a Principal Diagonal. In the language of projection, the criterion we wish to establish is this:
It is always possible to construct a projection in which two of the Principal Diagonals are corresponding rays from projective centers, and the the third Diagonal as the axis of perspective upon which the two rays intersect. That is, the three Principal Diagonals meet at a single point.
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Brianchon's Hexagram Theorem
This is problem #62 in Heinrich Dorrie's 100 Great Problems of Elementary Mathematics
More: http://tube.geogebra.org/material/show/id/73813
Used in: Conic from Five Tangents -- Drawing Solution: http://tube.geogebra.org/material/show/id/337589