Pasch's theorem

Pasch's theorem is an important idea in geometry. It was stated in 1882 by the German mathematician Moritz Pasch. This theorem deals with shapes and lines in plane geometry, and it shows something that cannot be proven using the old rules made by Euclid's postulates.

The theorem helps us understand how points and lines behave when they form triangles. It gives a way to think about when certain points lie inside or outside of shapes, which is useful in many areas of math.

Pasch's theorem is different from Pasch's axiom, which is another rule about lines and triangles. Both ideas help mathematicians study the relationships between points, lines, and shapes in a more careful way.

Statement

Pasch's theorem is a rule in plane geometry. It says that if you have four points, a, b, c, and d, all on the same straight line, and you know that point b is between a and c, and point c is between b and d, then point b must also be between a and d. This helps us understand how points are arranged on a line.

Hilbert's use of Pasch's theorem

David Hilbert included Pasch's theorem as one of his basic rules when he wrote about Euclidean geometry in his famous book The Foundations of Geometry in 1899. Later, in 1902, a mathematician named E. H. Moore showed that this rule wasn’t really needed as a basic rule, so Hilbert changed his book to treat Pasch's theorem as something that can be proven instead. Because of this, Pasch's theorem is sometimes called Hilbert's discarded axiom.

There is another rule called Pasch's axiom, which Hilbert kept as a basic rule in his work.