Napoleon's theorem
Imagine a triangle drawn on paper with sides of any length. On each side, construct an equilateral triangle pointing outward. Mark the center point of each new shape. Connect these three centers together. The resulting figure is always an equilateral triangle itself. This geometric fact holds true regardless of the original triangle's shape or size. Mathematicians call this result Napoleon's theorem. It applies whether the small triangles point inward or outward from the main shape. The difference in area between the outer and inner versions equals the area of the starting triangle.
The theorem bears the name of Napoleon Bonaparte who lived from 1769 to 1821. Howard Eves suggests the discovery actually belongs to Lorenzo Mascheroni who was his friend and adviser. Mascheroni reportedly let the Emperor claim credit for the work. A question appeared in W. Rutherford's 1825 publication called The Ladies' Diary four years after Napoleon died. Examination papers from the University of Dublin in October 1820 contain similar problems before the French emperor passed away in May 1821. Chambers's Encyclopaedia mentioned Napoleon by name in connection with this geometry as early as 1867. James Thomson published a textbook in 1834 that included the problem without proof. He later added a demonstration provided by Adam D. Glasgow of Belfast in 1837.
A quick way to see the equilateral nature involves rotating points around specific centers. A clockwise rotation of 30 degrees combined with a homothety ratio transforms one segment into another. A counterclockwise rotation of 30 degrees achieves the same transformation on the opposite side. These spiral similarities imply the angle between them must be exactly 60 degrees. Mathematicians have developed many different approaches to prove this statement. H.S.M. Coxeter and Samuel L. Greitzer wrote about synthetic coordinate-free proofs in their 1967 book Geometry Revisited. Other methods include trigonometric calculations, symmetry-based arguments, and complex number systems. Visual demonstrations exist at Cut-the-Knot showing how two rotations create the result.
The area of an inner Napoleon triangle depends on the side lengths of the original shape. Weitzenböck's inequality states equality occurs only when the starting triangle is already equilateral. Algebraically the inner triangle is retrograde so its signed area becomes negative of that expression. The outer Napoleon triangle has a specific area formula derived from the side lengths. Each side of the outer version measures a length calculated from those same sides. Analytically the area of any equilateral triangle equals the square of its side times a constant factor. This relationship connects the geometric construction directly to algebraic properties of the numbers involved.
Centers of both inner and outer Napoleon triangles coincide with the centroid of the original figure. P.G. Tait noted this property in Chambers's Encyclopaedia published in 1867. J.U. Hillhouse served as Mathematical Tutor at the University of Edinburgh during that period. Tait treated the problem in Section 189e of An Elementary Treatise on Quaternions released by Clarendon Press. He concluded that mean points of outwardly erected equilateral triangles form an equilateral triangle. The discussion appeared in subsequent editions from 1873 and 1890 alongside Philip Kelland. Perpendiculars erected at midpoints of sides prove the coincidence of these central points.
The Petr-Douglas-Neumann theorem extends the concept to polygons with more than three sides. Isosceles triangles with specific apex angles are erected on sides of an arbitrary n-gon. Repeating this process with different values forms a regular polygon whose centroid matches the original. A.J. Barlotti proved centers of regular n-gons constructed over sides form another regular n-gon if the shape is an affine image. M. de Villiers and colleagues described a hexagon variation where centroids form equilateral triangles. Dao Than Oai generalized the result further using alternate sides of a hexagon ABCDEF. These extensions show how a simple geometric observation can expand into complex mathematical structures.
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Common questions
What is Napoleon's theorem and how does it work?
Napoleon's theorem states that connecting the centers of equilateral triangles built on the sides of any triangle always results in an equilateral triangle. This geometric fact holds true regardless of the original triangle's shape or size whether the small triangles point inward or outward.
Who discovered Napoleon's theorem and when was it first published?
The theorem bears the name of Napoleon Bonaparte who lived from 1769 to 1821 but Howard Eves suggests the discovery actually belongs to Lorenzo Mascheroni. A question appeared in W. Rutherford's 1825 publication called The Ladies' Diary four years after Napoleon died while examination papers from the University of Dublin in October 1820 contain similar problems before the French emperor passed away in May 1821.
How can you prove Napoleon's theorem using rotations?
A clockwise rotation of 30 degrees combined with a homothety ratio transforms one segment into another while a counterclockwise rotation of 30 degrees achieves the same transformation on the opposite side. These spiral similarities imply the angle between them must be exactly 60 degrees which proves the equilateral nature of the resulting figure.
What is the area relationship between inner and outer Napoleon triangles?
The difference in area between the outer and inner versions equals the area of the starting triangle. Weitzenböck's inequality states equality occurs only when the starting triangle is already equilateral and algebraically the inner triangle is retrograde so its signed area becomes negative of that expression.
Where do the centers of Napoleon triangles coincide with other points?
Centers of both inner and outer Napoleon triangles coincide with the centroid of the original figure as noted by P.G. Tait in Chambers's Encyclopaedia published in 1867. Perpendiculars erected at midpoints of sides prove the coincidence of these central points and mean points of outwardly erected equilateral triangles form an equilateral triangle.