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65.6 The glass rod problem

As an aside, afterwards, he showed this elegant solution. The physical condition for success is that the longest piece must be less than or equal to half the original length (if it is longer the other two pieces cannot reach from one end to the other). Consider an equilateral triangle whose height is the length of the rod (say 1). It is easily proved that the sum of the lengths of the perpendiculars from the three sides to any interior point is 1. Further, given any triple of non-negative numbers whose sum is 1, there is just one interior point whose distances from the three sides (in some prescribed order) are precisely those three numbers. Hence there is a 1:1 correspondence between the interior points of the triangle and the possible lengths of the three pieces of the broken rod (see (a) in the figure).