Related Results
This theorem is a generalization of Pappus's (hexagon) theorem – Pappus's theorem is the special case of a degenerate conic of two lines. Pascal's theorem is the polar reciprocal and projective dual of Brianchon's theorem. It was discovered by Blaise Pascal in 1639 when he was 16 years old and published the following year as a broadside titled "Essay povr les coniqves. Par B. P.". Pascal deduced over four hundred corollaries from this theorem.
A degenerate case of Pascal's Theorem (four points) is interesting; given points ABCD on a conic Γ, the intersection of alternate sides, AB ∩ CD, BC ∩ DA, together with the intersection of tangents at opposite vertices (A, C) and (B, D) are collinear in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon' and the corresponding Pascal Line sharing either degenerate intersection. This can be proven independently using a property of pole-polar. If the conic is a circle, then another degenerate case tells us that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the Gergonne triangle, are collinear.
Six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic.
The converse is the Braikenridge–Maclaurin theorem, named for 18th century British mathematicians William Braikenridge and Colin Maclaurin (Mills 1984), which states that if the 3 intersection points of the lines through three sides of a hexagon lie on a line, then the 6 vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's theorem. The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a synthetic construction of the conic defined by five points, by varying the sixth point.
The theorem was generalized by Möbius in 1847, as follows: suppose a polygon with 4n + 2 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in 2n + 1 points. Then if 2n of those points lie on a common line, the last point will be on that line, too.
Read more about this topic: Pascal's Theorem
Famous quotes containing the words related and/or results:
“Becoming responsible adults is no longer a matter of whether children hang up their pajamas or put dirty towels in the hamper, but whether they care about themselves and others—and whether they see everyday chores as related to how we treat this planet.”
—Eda Le Shan (20th century)
“If family communication is good, parents can pick up the signs of stress in children and talk about it before it results in some crisis. If family communication is bad, not only will parents be insensitive to potential crises, but the poor communication will contribute to problems in the family.”
—Donald C. Medeiros (20th century)