scholarly journals Another generalisation of Napoleon's theorem

2009 ◽  
pp. 137-145
Author(s):  
G.C. Shephard
Keyword(s):  
Napoleon’S Theorem

An Analogue of Napoleon’s Theorem in the Hyperbolic Plane

2001 ◽  
Vol 44 (3) ◽  
pp. 292-297 ◽  
Author(s):  
Angela McKay
Keyword(s):  
Equilateral Triangle ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Equilateral Triangles ◽  
Napoleon’S Theorem

AbstractThere is a theorem, usually attributed to Napoleon, which states that if one takes any triangle in the Euclidean Plane, constructs equilateral triangles on each of its sides, and connects the midpoints of the three equilateral triangles, one will obtain an equilateral triangle. We consider an analogue of this problem in the hyperbolic plane.

Napoleon’s quasigroups

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Vedran Krčadinac
Keyword(s):  
Plane Geometry ◽  
Equilateral Triangles ◽  
Napoleon’S Theorem

AbstractNapoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab·b)(b·ba) = b. In works by V. Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids.

Plane polygons revisited

1982 ◽  
Vol 19 (A) ◽  
pp. 113-122 ◽  
Author(s):  
B. H. Neumann
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Finite Dimensional ◽  
Electrical Engineers ◽  
Equilateral Triangles ◽  
Arbitrary Plane ◽  
Arbitrary Triangle ◽  
Napoleon’S Theorem ◽  
Current Systems

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

Converses of Napoleon's Theorem

1992 ◽  
Vol 99 (4) ◽  
pp. 339-351 ◽  
Author(s):  
John E. Wetzel
Keyword(s):  
Napoleon’S Theorem
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