Pompeiu's theorem is a result of
plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, discovered by the Romanian mathematician
Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following:
:''Given an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
ABC in the plane, and a point P in the plane of the triangle ABC, the lengths PA, PB, and PC form the sides of a (maybe, degenerate) triangle.''
[Titu Andreescu, Razvan Gelca: ''Mathematical Olympiad Challenges''. Springer, 2008, , pp]
4-5
/ref>
The proof is quick. Consider a rotation of 60° about the point ''B''. Assume ''A'' maps to ''C'', and ''P'' maps to ''P'' '. Then , and . Hence triangle ''PBP'' ' is equilateral and . Then . Thus, triangle ''PCP'' ' has sides equal to ''PA'', ''PB'', and ''PC'' and the proof by construction
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existenc ...
is complete (see drawing).[Jozsef Sandor]
''On the Geometry of Equilateral Triangles''
Forum Geometricorum, Volume 5 (2005), pp. 107–117
Further investigations reveal that if ''P'' is not in the interior of the triangle, but rather on the circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
, then ''PA'', ''PB'', ''PC'' form a degenerate triangle, with the largest being equal to the sum of the others, this observation is also known as Van Schooten's theorem
Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles. It states:
:''For an equilateral triangle \triangle ABC with a point P on its circumcircle the length of longest of th ...
.
Generally, by the point ''P'' and the lengths to the vertices of the equilateral triangle - ''PA'', ''PB'', and ''PC'' two equilateral triangles ( the larger and the smaller) with sides and are defined:
:.
The symbol △ denotes the area of the triangle whose sides have lengths ''PA'', ''PB'', ''PC''.[Mamuka Meskhishvili]
'' Two Non-Congruent Regular Polygons Having Vertices at the Same Distances from the Point''
International Journal of Geometry, Volume 12 (2023), pp. 35–45
Pompeiu published the theorem in 1936, however August Ferdinand Möbius
August Ferdinand Möbius (, ; ; 17 November 1790 – 26 September 1868) was a German mathematician and theoretical astronomer.
Early life and education
Möbius was born in Schulpforta, Electorate of Saxony, and was descended on hi ...
had published a more general theorem about four points in the Euclidean plane already in 1852. In this paper Möbius also derived the statement of Pompeiu's theorem explicitly as a special case of his more general theorem. For this reason the theorem is also known as the ''Möbius-Pompeiu theorem''.[D. MITRINOVIĆ, J. PEČARIĆ, J., V. VOLENEC: '' History, Variations and Generalizations of the Möbius-Neuberg theorem and the Möbius-Ponpeiu''. Bulletin Mathématique De La Société Des Sciences Mathématiques De La République Socialiste De Roumanie, 31 (79), no. 1, 1987, pp. 25–38]
JSTOR
External links
Pompeiu's theorem
at cut-the-knot.org
Notes
Elementary geometry
Theorems about equilateral triangles
Theorems about triangles and circles
Articles containing proofs