Suppose that the points A,B,C,D,E,F lie on a non-degenerate conic.
Suppose also that AC, DB meet in X, CF, BE in Y and FD, EA in Z.
Then X, Y and Z are collinear.
The figure was drawn with Cabri and displayed using CabriJava.
See the Technical Notes below for more details.
(1) We can think of X,Y,Z as the intersections of opposite sides of the hexagon ACFDBE.
(2) The order of the points round the conic is immaterial. In the figure we initially have
the points in the cyclic order ABCDEF.
(3) The theorem holds for any non-degenerate conic. The figure shows an ellipse.
(4) If the conic degenerates into a pair of lines, and A,B,F lie on one line and C,D,E on the
other, the result still holds. This is Pappus's Theorem.
(5) A proof, with necessary prerequisites, can be found here.
(6) The converse of Pascal's Theorem is valid, and has a useful consequence.
Blaise Pascal discovered this theorem in 1640 - at the age of 16.
Cabri is a dynamic geometry program. It allows the construction of geometrical figures.
It is dynamic since the geometrical objects may be moved. When the point F above is moved,
the point Y moves so that it is the intersection of CF and BE. The line XY is redrawn. The point
Z moves so it is the intersection of FD and EA. Thus, everything that depends on F is updated.
CabriJava is a piece of Java code which displays Cabri figures on the Web. The viewer may
interact with the figure, as in Cabri, but cannot create new objects.
|Main Geometry Page||More Cabri Java|