Scholarlink

<p>In the well-known Napoleon’s theorem, equilateral triangles are built either all inwards or all outwards on the sides of a triangle. The centers of these triangles then form an equilateral triangle. Thébault’s theorem makes an analogous statement for parallelograms: If one erects squares on the sides (all inwards or all outwards), their centers form a square. Varignon’s theorem applies to general quadrilaterals: The centers of the sides form a parallelogram. We show the following for general quadrilaterals: Erect directly similar triangles on the sides, <i>alternately</i> inwards and outwards. Then the vertices, the centers of gravity, the orthocenters, the centers of the inscribed circle, or any other centers of these triangles always form a parallelogram. We also explore some of the nice geometric properties of these parallelograms.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

When Varignon meets Napoleon

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年4月8日

查看全文
  • Norbert Hungerbühler

摘要

In the well-known Napoleon’s theorem, equilateral triangles are built either all inwards or all outwards on the sides of a triangle. The centers of these triangles then form an equilateral triangle. Thébault’s theorem makes an analogous statement for parallelograms: If one erects squares on the sides (all inwards or all outwards), their centers form a square. Varignon’s theorem applies to general quadrilaterals: The centers of the sides form a parallelogram. We show the following for general quadrilaterals: Erect directly similar triangles on the sides, alternately inwards and outwards. Then the vertices, the centers of gravity, the orthocenters, the centers of the inscribed circle, or any other centers of these triangles always form a parallelogram. We also explore some of the nice geometric properties of these parallelograms.