寻找梅尔基奥尔的普通点
In Search of Melchior's Ordinary Points
AI总结：
针对西尔维斯特提出的平面点集普通直线问题，本文给出了梅尔基奥尔对偶问题中全部3个普通点的简单可视化证明。
AI中文摘要：
1893年，詹姆斯·约瑟夫·西尔维斯特提出如下问题：给定平面上n个不全共线的点，是否必存在一条由其中两点确定且不经过其他任何点的直线？1940年，埃伯哈德·梅尔基奥尔在实射影平面中研究了等价的对偶问题：给定射影平面上n条不共点的直线，是否必存在一个恰好由两条直线相交得到的点？这类交点被称为普通点。梅尔基奥尔通过巧妙的双重计数论证，得出这类点至少存在3个。如今被称为西尔维斯特-加拉伊定理的结论是必存在至少一个普通点，已有诸多简单的“可视化”证明，一个自然的问题是：是否存在简单的可视化证明可得到梅尔基奥尔的全部3个普通点？本文提供了这样一个证明。
英文摘要：
In 1893, James Joseph Sylvester posed the following problem: given n points in the plane, not all collinear, must there be a line determined by two of the points that does not pass through any of the other points? In 1940, Eberhard Melchior studied the equivalent dual problem in the projective plane: given a set of n lines in the (real) projective plane, not all passing through a common point, must there be a point where exactly two of the lines intersect? Such a point of intersection is called an *ordinary point*. Via a clever double-counting argument, Melchior found that in fact there must be at least three such points. Given the many simple "visual" proofs of what is today known as the Sylvester-Gallai Theorem --- the theorem that states there must be at least one ordinary point --- a natural question is whether there is a simple visual proof that recovers all three of Melchior's ordinary points. This paper provides such a proof.
