Abstract

As is well-known, numerical experiments show that Napoleon’s Theorem for planar triangles does not extend to a similar statement for triangles on the unit sphere \(S^2\) . Spherical triangles for which an extension of Napoleon’s Theorem holds are called Napoleonic, and until now the only known examples have been equilateral. In this paper we determine all Napoleonic spherical triangles, including a class corresponding to points on a 2-dimensional ellipsoid, whose Napoleonisations are all congruent. Other new classes of examples are also found, according to different versions of Napoleon’s Theorem for the sphere. The classification follows from successive simplifications of a complicated original algebraic condition, exploiting geometric symmetries and algebraic factorisations.

中文翻译：

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球体上的拿破仑三角形

摘要

众所周知，数值实验表明平面三角形的拿破仑定理并不能扩展到单位球体\(S^2\)上的三角形的类似陈述。拿破仑定理的扩展成立的球面三角形称为拿破仑三角形，迄今为止唯一已知的例子是等边三角形。在本文中，我们确定了所有拿破仑球面三角形，包括对应于二维椭球上的点的类，其拿破仑化都是全等的。根据拿破仑球体定理的不同版本，还发现了其他新类别的示例。该分类是利用几何对称性和代数分解对复杂的原始代数条件进行连续简化而得出的。