[b]Proposition:[/b] [i]To Demonstrate that the opposite vertex lines of a hexagram circumscribed about a conic section pass through a point.[/i]
[b]Converse:[/b] [i]If the opposite vertex lines of hexagram pass through a point, the sides of the hexagram form tangents of a conic section.[/i]
Demonstration of the Brianchon point. Proof of this theorem is quite a challenge! Here is a brief outline of Dorrie's proof, relying heavily on projection theorems of Steiner and Desargues. For convenience, I refer to a line joining opposite vertices a [i]Principal Diagonal[/i]
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[*]We first establish
[math]\;\;\; [/math][b]a)[/b] Any tangent to a circle cuts any two other tangents in projective ranges of points
[math]\;\;\; [/math][b]b) [/b] We can project a circle to any conic section, and so [b]a.[/b] holds for conic sections.
Then, in words,
[*] [i]we can always 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[/i].
That is, the three Principal Diagonals meet at a single point. Q.E.D.
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[b]Brianchon's Hexagram Theorem[/b]
This is problem #62 in Heinrich Dorrie's [i]100 Great Problems of Elementary Mathematics[/i]
More: [url]http://tube.geogebra.org/material/show/id/73813[/url]
Used in: Conic from Five Tangents -- Drawing Solution: [url]http://tube.geogebra.org/material/show/id/337589[/url]
