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This morning, in Italy, there was the national exam of mathematics for students of high schools. One of the exercises asked to solve Heron's problem: given a straight line and two points lying on the same side of the line, find the best path (= the path of minimal length) that connects them and touches the straight line.

As a mathematician I (probably) know the answer. However, every solution published by newspapers assumes that the optimal path is made of two segments, i.e. the solution must be found among piecewise affine curves. This is true, but can such a solution be accepted as correct? Actually, the problem seems rather hard, if no regularity assumption on the class of admissible paths is made.

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    Consider the segment of the path from one of the given points to the straight line. If the segment is not straight, you can replace it with a straight line segment while keeping its endpoints fixed. This reduces the length of the path, so the original path cannot be optimal.2012-06-21
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    An intuitive solution: make symmetry of one point respect to the straight line. The minimum distance from this symmetric point to the other point is a straight line. Then the curve with minimum length between the two points is comprised by two segments.2012-06-21
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    I think one is allowed to assume without proof that the shortest connection of two points is a segment. Given that, the well known solution is easy to establish.2012-06-21
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    There was some discussion of this question at http://math.stackexchange.com/questions/153219/calculating-the-shortest-possible-distance-between-points/153227#1532272012-06-22
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    Thank you for your comments. My question was mainly about assessment: should a **complete** solution contain the proof the shortest path is piecewise affine? It is a rather elementary fact, but it is also the most interesting part of the problem, since the rest is elementary geometry.2012-06-22
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    I guess this question was indirectly answered by some of the commments. The fact that a straight line is the shortest path between two points in the plane could be taken for granted. If the solution builds on that fact (e.g. by using reflection across the line) I'd accept. Otherwise I'd ask for additional reasoning.2012-06-24
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    I may agree. But I think that this should be clearly written by the students.2012-06-24

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