Möbius-Pompeiu's theorem

Pompeiu triangle

{\ displaystyle \ triangle CP'P}

The set of Mobius Pompeiu , named after Dimitrie Pompeiu and August Ferdinand Möbius , describes a property of equilateral triangles . It says that the three connecting lines that any point in the plane forms with the corner points of an equilateral triangle always satisfy the triangle inequality . This means that the sum of the lengths of two connecting links is always greater than or equal to the length of the third connecting link. This means that a triangle can always be constructed, the side lengths of which correspond to the lengths of the connecting lines; such a triangle is called a Pompeiu triangle .

If the point lies on the circumference of the equilateral triangle, the associated Pompeiu triangle is only a degenerate triangle (line) and the sum of the lengths of the two shorter connecting lines corresponds exactly to the length of the third connecting line. This special case is also known as Van Schooten's theorem.

A simple geometrical proof of the theorem results if one rotates the initial configuration around one of the corner points of the equilateral triangle. If you turn the point and the connecting lines and around the corner point of the equilateral triangle clockwise and designate the image with , the triangle is equilateral. The lengths of the sides of the triangle thus correspond exactly to the lengths of the connecting lines. ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle P}$ ${\ displaystyle PA}$ ${\ displaystyle PB}$ ${\ displaystyle B}$ ${\ displaystyle \ triangle ABC}$ ${\ displaystyle 60 ^ {\ circ}}$ ${\ displaystyle P}$ ${\ displaystyle P '}$ ${\ displaystyle \ triangle PBP '}$ ${\ displaystyle \ triangle CP'P}$

The sentence, which is often referred to in the literature as Pompey's sentence , was published by Pompeiu in 1936. However, as early as 1852 Möbius published a more general theorem on 4 points in the plane, which contains Pompey's theorem as a special case.

literature

Jozsef Sandor: On the Geometry of Equilateral Triangles . In: Forum Geometricorum. Volume 5, 2005, pp. 107-117.
DS Mitrinović, JE Pečarić, V. Volenec: History, Variations and Generalizations of the Möbius-Neuberg theorem and the Möbius-Ponpeiu. In: Bulletin Mathématique De La Société Des Sciences Mathématiques De La République Socialiste De Roumanie. Volume 31 (79), No. 1, 1987, pp. 25-38. ( JSTOR 43681294 )
Titu Andreescu, Razvan Gelca: Mathematical Olympiad Challenges . Springer, 2008, ISBN 978-0-8176-4611-0 , pp. 4-5. (books.google.de)
FA Möbius: About a method to get from relations that belong to longimetry to corresponding sets of planimetry. In: Journal for pure and applied mathematics. Volume 1856, Issue 52, pp. 229-242, doi: 10.1515 / crll.1856.52.229 .

Web links

Sentence by Pompeiu on cut-the-knot.org

Individual evidence

↑ ^a ^b Titu Andreescu, Razvan Gelca: Mathematical Olympiad Challenges . Springer, 2008, ISBN 978-0-8176-4611-0 , pp. 4-5. (books.google.de)
^ ^A ^b Jozsef Sandor: On the Geometry of Equilateral Triangles . In: Forum Geometricorum. Volume 5, 2005, pp. 107-117.
↑ DS Mitrinović, JE Pečarič, V. Volenec: History, Variations and Generalizations of the Möbius Neuberg theorem and the Mobius Ponpeiu. In: Bulletin Mathématique De La Société Des Sciences Mathématiques De La République Socialiste De Roumanie. Volume 31 (79), No. 1, 1987, pp. 25-38 ( JSTOR 43681294 )

[Titu-1] Titu Andreescu, Razvan Gelca: Mathematical Olympiad Challenges . Springer, 2008, ISBN 978-0-8176-4611-0 , pp. 4-5. (books.google.de)

[Sandor-2] A ^b Jozsef Sandor: On the Geometry of Equilateral Triangles . In: Forum Geometricorum. Volume 5, 2005, pp. 107-117.

[3] DS Mitrinović, JE Pečarič, V. Volenec: History, Variations and Generalizations of the Möbius Neuberg theorem and the Mobius Ponpeiu. In: Bulletin Mathématique De La Société Des Sciences Mathématiques De La République Socialiste De Roumanie. Volume 31 (79), No. 1, 1987, pp. 25-38 ( JSTOR 43681294 )