Napoleon's theorem
Encyclopedia
In mathematics
, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle
, either all outward, or all inward, the centre
s of those equilateral
triangle
s themselves form an equilateral triangle.
The triangle thus formed is called the Napoleon triangle (inner and outer). The difference in area of these two triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon (1769–1821). However, it may just date back to W. Rutherford's 1825 publication The Ladies' Diary
, four years after the French emperor's death.
A quick way to see that the triangle LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and an homothety of ratio √3 with the same center and that and LN also becomes CZ after a counterclockwise rotation of 30° around B and an homotecy of ratio √3 with the same center. the respective spiral similarities A(√3,-30°) and B(√3,30°). That implies MN = LN and the angle between them must be 60°.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
, either all outward, or all inward, the centre
Centre (geometry)
In geometry, the centre of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries.-Circles:...
s of those equilateral
Equilateral
In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...
triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s themselves form an equilateral triangle.
The triangle thus formed is called the Napoleon triangle (inner and outer). The difference in area of these two triangles equals the area of the original triangle.
The theorem is often attributed to Napoleon (1769–1821). However, it may just date back to W. Rutherford's 1825 publication The Ladies' Diary
The Ladies' Diary
The Ladies' Diary: or, Woman's Almanack appeared annually in London from 1704 to 1841. It featured material relating to calendars etc. including sunrise and sunset times and phases of the moon, as well as important dates , and a chronology of remarkable events.The subtitle indicated its serious...
, four years after the French emperor's death.
A quick way to see that the triangle LMN is equilateral is to observe that MN becomes CZ under a clockwise rotation of 30° around A and an homothety of ratio √3 with the same center and that and LN also becomes CZ after a counterclockwise rotation of 30° around B and an homotecy of ratio √3 with the same center. the respective spiral similarities A(√3,-30°) and B(√3,30°). That implies MN = LN and the angle between them must be 60°.
External links
- Napoleon's Theorem and Generalizations
- To see the construction
- Napoleon's Theorem by Jay Warendorff, The Wolfram Demonstrations Project.
- Napoleon's Theorem and some generalizations, variations & converses at Dynamic Geometry Sketches