The Fermat-Steiner Problem
The Fermat-Steiner Problem
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DOI：
10.1080/00029890.2002.11919871
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发表时间：
2002-05
期刊：
影响因子：
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通讯作者：
S. Gueron;Ran Tessler
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文献类型：
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作者：
S. Gueron;Ran Tessler
这个问题的原始处理方法只考虑了锐角三角形，P必须是一个内点。因此，关于P相对于三角形的相对位置的讨论并没有出现。让我们首先对这个相对位置做一些简单的观察：P不能位于ABC之外，因为在这种情况下，"最近点投影"，称之为P ′，给出G（P ′）<G（P）。根据距离函数的凸性，如果P在ABC内，则它是唯一的。而且，很明显，如果P位于ABC的一边，它一定是一个顶点。总结：我们的最小化候选是ABC的内部点和顶点。Courant和Robbins [4，pp. 354 - 359]属性这个问题，著名的18世纪世纪瑞士几何雅各布施泰纳谁研究它，可能独立的早期参考，我们今天知道。施泰纳推导出了一个涵盖所有情况的系统解决方案[9，第。24 - 35]。他的答案可以概括如下：如果ABC的所有角都小于120 °，那么P是ABC内部的一点，从这个点可以看到ABC的边的角度为120 °。如果ABC的一个角至少是120度，那么P就是这个角的顶点。然而，历史表明，我们问题的根源及其解决方案比施泰纳的分析要早得多。正是费马（1601 - 1665）向托里切利（1608 - 1647）提出了这个问题。托里切利解决了这个问题，并将其沿着交给了他的学生维维亚尼（Viviani，1622 - 1703年），后者于1659年发表了他自己和托里切利的解决方案。最早的书面讨论这个问题，我们能够跟踪发现在卡瓦利耶里的书从1647年;见[1]，[13，页。443 - 444]。托里切利的解决方案使用了我们今天所说的维维亚尼定理：任何内部点M到等边三角形各边的距离之和等于三角形的高度。为了解决这个问题，托里切利考虑了三角形内的一个点P，从这个点可以看到AB，BC，CA的边在120度角处。他构造了一个辅助三角形，其边分别通过A，B，C，并垂直于P A，P B，PC。这个辅助三角形是等边的（见图1），应用Viviani定理解决了这个问题。
The original treatment of the problem considered only triangles with acute angles, for which P must be an interior point. Therefore, a discussion of the relative position of P with respect to the triangle did not emerge. Let us first make some simple observations about this relative position: P cannot lie outside ABC because in this case the “nearest point projection”, call it P ′, gives G(P ′) < G(P). By the convexity of the distance function it follows that if P is inside ABC , it is unique. Also, it is clear that if P lies on a side of ABC , it must be a vertex. To conclude: our minimizing candidates are the interior points and the vertices of ABC . Courant and Robbins [4, pp. 354–359] attribute this problem to the famous 18th century Swiss geometer Jacob Steiner who studied it, probably independently of the earlier references that we are aware of today. Steiner derived a systematic solution that covers all cases [9, pp. 24–35]. His answer can be summarized as follows: If all of the angles of ABC are less than 120◦, then P is the point inside ABC from which its sides are seen at the angle 120◦. If one angle of ABC is at least 120◦, then P is the vertex at this angle. History, however, indicates that the roots of our problem and its solutions are much earlier than Steiner’s analysis. It was Fermat (1601–1665) who proposed this problem to Torricelli (1608–1647). Torricelli solved the problem and passed it along to his student Viviani (1622–1703), who published his own and Torricelli’s solution in 1659. The earliest written discussion of this problem that we were able to trace is found in Cavallieri’s book from 1647; see [1], [13, pp. 443–444]. Torricelli’s solution uses what we call today Viviani’s Theorem: the sum of the distances of any interior point M from the sides of an equilateral triangle equals the altitude of the triangle. To solve the problem, Torricelli considered a point P inside the triangle from which the sides AB, BC , C A are seen at the angle 120◦. He constructed an auxiliary triangle whose sides pass through A, B, C and are perpendicular to P A, P B, PC , respectively. This auxiliary triangle is equilateral (see Figure 1), and applying Viviani’s Theorem solves the problem.