# Brianchon's Hexagram Theorem

- Author:
- Ryan Hirst

**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. _________________**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