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Pascalā€™s theorem
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Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear. Figure 7: Construction for Pappus' theorem (see text). Pascal's hexagon
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Pascalā€™s theorem

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Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.
Pascal's projective theorem

The 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.

Pascal's theorem is a fundamental concept in projective geometry, a branch of mathematics that studies geometric figures and their mappings. The theorem, discovered by Blaise Pascal, deals with the properties of points and lines, particularly when projected onto another surface. It states that if six points lie on a circle, the three intersection points of the lines connecting these points in a specific order are collinear (lie on the same straight line). This theorem is significant because it highlights invariants under projective mappings, meaning certain properties remain unchanged even when figures are transformed through projection. It demonstrates that lines map to lines, and collinearity is preserved. Pascal's theorem can be generalized to any conic section, not just circles, and it unifies different types of conics from a projective viewpoint.12

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5 Britannica Sources
  1. projective geometry
  2. binomial theorem
  3. Desarguesā€™s theorem
  4. Blaise Pascal: Top Questions
  5. Pappusā€™s theorem

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