Napoleon triangle

Napoleon triangle , named after the French general and emperor Napoléon Bonaparte , is a term used in triangular geometry .

definition

Three equilateral triangles are drawn over the sides of a given triangle ABC and the geometric centers of gravity (area centers of gravity) are entered in each of these . The Napoleon triangle is created by connecting these focal points.

If the equilateral triangles are laid out facing outwards, the center of gravity connection results in the outer Napoleon triangle; if the equilateral triangles are laid out inwards, the inner Napoleon triangle is obtained.

The Napoleon triangle is - regardless of the shape of the original triangle - always equilateral.

It is not clear whether this sentence was actually found by Napoleon .

properties

The center of gravity of the given triangle coincides with the center of gravity of the Outer Napoleon Triangle and the center of gravity of the Inner Napoleon Triangle. If one forms the difference between the areas of the outer Napoleon triangle and the inner Napoleon triangle, then one obtains the area of ​​the given triangle.

generalization

If the three equilateral triangles are replaced by similar isosceles triangles in the definition , one speaks of a Kiepert triangle .

See also

• Excellent points in the triangle
• Napoleon point
• Van Aubel's theorem

literature

• HSM Coxeter , SL Greitzer: Timeless geometry . Klett, Stuttgart 1983

Web links

• Ulf von Rauchhaupt: Napoleon's Theorem . FAZ.net , August 15, 2019
• Napoleon triangle - a visualization with the dynamic geometry program GeoGebra
• Eric W. Weisstein : Inner Napoleon Triangle . In: MathWorld (English).
• Eric W. Weisstein : Outer Napoleon Triangle . In: MathWorld (English).