geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, Brianchon's theorem is a theorem stating that when a

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

circumscribed In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

around a

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

, its principal

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

s (those connecting opposite vertices) meet in a single point. It is named after

Charles Julien Brianchon Charles Julien Brianchon (19 December 1783 – 29 April 1864) was a French mathematician and chemist. Life He entered into the École Polytechnique in 1804 at the age of eighteen, and studied under Monge, graduating first in his class in 1808 ...

(1783–1864).

Formal statement

Let

P_1P_2P_3P_4P_5P_6

be a

formed by six

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

s of a

. Then lines

\overline,\; \overline,\; \overline

(extended diagonals each connecting opposite vertices) intersect at a single

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

B

, the Brianchon point.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

Connection to Pascal's theorem

The polar reciprocal and projective dual of this theorem give

Pascal's theorem In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by ...

Degenerations

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 In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, t ...

s of triangles. From a projective point of view the two triangles

P_1P_3P_5

and

P_2P_4P_6

lie perspectively with center

B

. 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

affine plane In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are * Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...

and the

real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...

. However, its statement in the affine plane is in a sense less informative and more complicated than that in the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

. Consider, for example, five tangent lines to a

parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...

. These may be considered sides of a hexagon whose sixth side is the

line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The l ...

, 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 In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of ...

or reciprocation.

References