Editing Brianchon's theorem

{{short description|The 3  long diagonals of a hexagon tangent to a conic section meet in a single point}}
[[File:Brianchon-6-tangents.svg|300px|thumb|Brianchon's theorem]]
In [[geometry]], '''Brianchon's theorem''' is a theorem stating that when a [[hexagon]] is [[circumscribed]] around a [[conic section]], its principal [[diagonal]]s (those connecting opposite vertices) meet in a single point. It is named after [[Charles Julien Brianchon]] (1783–1864).

==Formal statement==
Let <math>P_1P_2P_3P_4P_5P_6</math> be a [[hexagon]] formed by six [[tangent line]]s of a [[conic section]]. Then lines  <math>\overline{P_1P_4},\; \overline{P_2P_5},\; \overline{P_3P_6}</math> (extended diagonals each connecting opposite vertices) intersect at a single [[Point (geometry)|point]] <math>B</math>, the '''Brianchon point'''.<ref name=WW>Whitworth, William Allen. ''Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions'', Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866).  http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books</ref>{{rp|p. 218}}<ref>{{cite book
 | author = Coxeter, H. S. M.
 | author-link = H. S. M. Coxeter
 | title = Projective Geometry
 | edition = 2nd
 | year = 1987
 | publisher = Springer-Verlag
 | isbn = 0-387-96532-7
 | pages = Theorem 9.15, p. 83
 | no-pp = true}}</ref>

==Connection to Pascal's theorem==
The [[polar reciprocation|polar reciprocal]] and [[projective dual]] of this theorem give [[Pascal's theorem]].

==Degenerations==
[[File:Brianchon-3-tangents.svg|300px|thumb|3-tangents degeneration of Brianchon's theorem]]
As for Pascal's theorem there exist ''degenerations'' for Brianchon's theorem, too: Let coincide two neighbored tangents. Their point of intersection becomes a point of the conic. In the diagram three pairs of neighbored tangents coincide. This procedure results in a statement on [[inellipse]]s of triangles. From a projective point of view the two triangles <math>P_1P_3P_5</math> and <math>P_2P_4P_6</math> lie perspectively with center <math>B</math>. That means there exists a central collineation, which maps the one onto the other triangle. But only in special cases this collineation is an affine scaling. For example for a Steiner inellipse, where the Brianchon point is the centroid.

==In the affine plane==
Brianchon's theorem is true in both the [[Euclidean plane|affine plane]] and the [[real projective plane]].  However, its statement in the affine plane is in a sense less informative and more complicated than that in the [[projective plane]].  Consider, for example, five tangent lines to a [[parabola]].  These may be considered sides of a hexagon whose sixth side is the [[line at infinity]], but there is no line at infinity in the affine plane.  In two instances, a line from a (non-existent) vertex to the opposite vertex would be a line ''parallel to'' one of the five tangent lines.  Brianchon's theorem stated only for the affine plane would therefore have to be stated differently in such a situation.

The projective dual of Brianchon's theorem has exceptions in the affine plane but not in the projective plane.

==Proof==
Brianchon's theorem can be proved by the idea of [[radical axis]] or reciprocation.
To prove it take an arbitrary length (MN) and carry it on the tangents starting from the contact points: PL = RJ = QH = MN etc. Draw circles a, b, c tangent to opposite sides of the hexagon at the created points (H,W), (J,V) and (L,Y) respectively. One sees easily that the concurring lines coincide with the radical axes ab, bc, ca resepectively, of the three circles taken in pairs. Thus O coincides with the radical center of these three circles.

The theorem takes particular forms in the case of circumscriptible pentagons e.g. when R and Q tend to coincide with F, a case where AFE is transformed to the tangent at F. Then, taking a further similar identification of points T,C and U, we obtain a corresponding theorem for quadrangles.

==See also==

* [[Seven circles theorem]]
* [[Pascal's theorem]]

==References==
{{reflist}}

[[Category:Conic sections]]
[[Category:Theorems in projective geometry]]
[[Category:Euclidean plane geometry]]
[[Category:Theorems about polygons]]
[[Category:Affine geometry]]