Navigators have been taught for centuries to estimate the location of their craft on a map from three lines of position, for redundancy. The three lines typically form a triangle, called a cocked hat. How is the location of the craft related to the triangle? For more than 80 years navigators have also been taught that, if each line of position is equally likely to pass to the right and to the left of the true location, then the likelihood that the craft is in the triangle is exactly 1/4. This is stated in numerous reputable sources, but was never stated or proved in a mathematically formal and rigorous fashion. In this paper we prove that the likelihood is indeed 1/4 if we assume that the lines of position always intersect pairwise. We also show that the result does not hold under weaker (and more reasonable) assumptions, and we prove a generalisation to $n$ lines.

中文翻译：

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三角帽：定理的正式陈述和证明

几个世纪以来，航海者一直被教导从地图上的三行位置估计他们的飞船的位置，以备冗余。这三条线通常形成一个三角形，称为三角帽。飞船的位置与三角形有什么关系？80 多年来，导航员们也被教导说，如果每条位置线都有可能通过真实位置的右侧和左侧，那么飞船在三角形中的可能性正好是 1/4。这在许多有信誉的来源中都有陈述，但从未以数学形式和严格的方式陈述或证明。在本文中，我们证明如果我们假设位置线总是成对相交，则可能性确实是 1/4。我们还表明，在较弱（和更合理）的假设下，结果并不成立，$n$线。