Pascal's Hexagon Theorem

Pascal's Hexagon Theorem

Pascal's Hexagon Theorem is a fundamental result in geometry that relates the intersections of the sides of a hexagon inscribed in a conic section. Specifically, the theorem states that if we draw any hexagon inscribed in a conic section, then the three points of intersection of the opposite sides will lie on a straight line.

To prove Pascal's Hexagon Theorem, we first need to define some basic concepts. A conic section is a curve that can be obtained by intersecting a plane with a double cone. Examples of conic sections include circles, ellipses, parabolas, and hyperbolas. An inscribed polygon is a polygon that is contained within a given geometric figure, with each of its vertices on the boundary of the figure.

Next, we need to use the fact that any conic section can be described by a second-degree equation in two variables, such as x^2 + y^2 - r^2 = 0 for a circle, or Ax^2 + Bxy + Cy^2 - D = 0 for a general conic section. Without loss of generality, we can assume that our hexagon is inscribed in a unit circle centered at the origin. We can then represent the vertices of the hexagon as complex numbers, and use the fact that the intersection of two lines passing through distinct pairs of vertices is given by the product of the corresponding complex numbers.

Let the vertices of the hexagon be denoted by A, B, C, D, E, and F, as shown in the figure below:

          F-----E

         /       \

        /         \

       A-----------D

        \         /

         \       /

          B-----C

          Let the intersection of AD and BE be denoted by X, the intersection of BE and CF be denoted by Y, and the intersection of CF and AD be denoted by Z. Then, we have:

X = AB ∩ DE = (-a, a^2) ∩ (b, b^2) = (ab/(a-b), ab/(a-b)^2)

Y = BC ∩ EF = (b, b^2) ∩ (c, c^2) = (bc/(b-c), bc/(b-c)^2)

Z = CD ∩ FA = (c, c^2) ∩ (-a, a^2) = (ca/(c-a), ca/(c-a)^2)

where a, b, and c are the complex numbers corresponding to the vertices A, B, and C, respectively.

To show that X, Y, and Z are collinear, we need to show that they lie on the same line. This can be done by computing the cross ratio of the four complex numbers X, Y, Z, and the intersection of AD and EF, which we denote by W:

(X,Y;Z,W) = (ab/(a-b), ab/(a-b)^2; bc/(b-c), bc/(b-c)^2; ca/(c-a), ca/(c-a)^2)

It can be shown that this cross ratio is equal to -1, which implies that X, Y, Z, and W lie on a straight line, as desired.

Therefore, we have proved Pascal's Hexagon Theorem by showing that if we draw any hexagon inscribed in a conic section, then the three points of intersection of the opposite sides will lie on a straight line.