Brianchon's theorem

Brianchon's theorem

The set of Brianchon , named after the French mathematician Charles Julien Brianchon (1783-1864), is a classic theorem of plane geometry .

• In a convex hexagon which has a non-degenerate conic circumscribes (d. E. All sides are tangents of the conic section), the diagonals will intersect at a point , the Brianchon point . ${\displaystyle P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}}$ ${\displaystyle P_{1}P_{4},P_{2}P_{5},P_{3}P_{6}}$ ${\displaystyle B}$

This is the dual version of Pascal's theorem .

3-tangent degeneration of Brianchon's theorem

As with Pascal's theorem, Brianchon's theorem also has degenerations . Neighboring tangents are allowed to coincide and their intersection becomes a conic intersection. In the example in the picture 3 pairs of tangents have coincided. This results in a statement about inellipses of triangles. From a projective point of view, one can also determine: The two triangles and are in perspective . That means there is a central collineation which maps one triangle onto the other triangle. Only in special cases is this central collineation also an affine mapping (stretching at a point), e.g. B. in a Steiner inellipse , both triangles are connected to one another via an extension at the center point, which is also Brianchon's point. ${\displaystyle P_{1}P_{3}P_{5}}$ ${\displaystyle P_{2}P_{4}P_{6}}$

literature

• Coxeter, HSM , and SL Greitzer: Timeless Geometry, Klett Stuttgart, 1983

Web links

• Brianchon's theorem on PlanetMath
• Brianchon's theorem on cut-the-knot