Math 2.0

Geometry Projective geometry and linear incidence geometry Theorem

Brianchon's theorem

Builds on Projective spaceProjective planeConic sectionTangent line

Statement

Work in the real projective plane ℙ2(ℝ)\mathbb{P}^2(\R), and let 𝒞\mathcal{C} be a nondegenerate conic. Let t1,…,t6t_1, \dots, t_6 be six distinct lines tangent to 𝒞\mathcal{C}. Put A=t1∩t2A = t_1 \cap t_2, B=t2∩t3B = t_2 \cap t_3, C=t3∩t4C = t_3 \cap t_4, D=t4∩t5D = t_4 \cap t_5, E=t5∩t6E = t_5 \cap t_6 and F=t6∩t1F = t_6 \cap t_1. Then the three lines ADAD, BEBE and CFCF pass through a common point.

Remarks

So ABCDEFABCDEF is a hexagon whose sides touch the conic, and the statement says that its three main diagonals are concurrent. No three tangent lines of a nondegenerate real conic pass through one point, which makes the six vertices distinct and the three diagonals well defined.

The theorem is the dual of Pascal's theorem. The tangent lines of a nondegenerate conic form a nondegenerate conic of the dual plane, and the dual of the statement "three points are collinear" is "three lines are concurrent". So the principle of duality applied to Pascal's theorem gives Brianchon's theorem.

Example: for a regular hexagon circumscribed about a circle, the three main diagonals pass through the centre of the circle.

Going deeper

For a student

Being the dual of Pascal's theorem, it is a classic illustration of projective duality. It is used to construct tangent lines of a conic from five given tangent lines with a straightedge. The same argument works over any field of characteristic different from 22. In characteristic 22 all tangent lines of a nondegenerate conic pass through one point, so the hypotheses of the statement cannot be met there.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Conic section, concept, no evidence levelConic sectionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionProjective plane, concept, no evidence levelProjective planeconceptProjective space, concept, no evidence levelProjective spaceconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionTangent line, concept, no evidence levelTangent lineconceptBrianchon's theorem, statement, no evidence levelBrianchon's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Conic section concept
  2. Limit point of a set definition
  3. Projective plane concept
  4. Projective space concept
  5. Limit of a function definition
  6. Derivative definition
  7. Tangent line concept
  8. Brianchon's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Brianchon%27s_theorem
About this page
Tag3THE3Q96KK
URIm2://3THE3Q96KK
Kindstatement, theorem
WritingW0 stub
MSC51A20, 51N15
StatusThis copy of the site has no build data. Run m2 status 3THE3Q96KK.
AboutConic section
UsesProjective space, Projective plane, Conic section, Tangent line
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