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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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I think it is sufficient common knowledge that the the shortest distance between two points is a straight line that it does not need to be proved (definition of straight line, triangle inequality etc.). So there are two-and-a-half obvious ways to find the answer:

  1. Start from the first given point, go straight to an as yet unknown point on the given line, go straight to a second unknown point on the given line, and then go straight to the second given point. Find the two unknown points which minimise the total of the three straight distances.

    1.5 In method 1, the total distance can clearly be reduced if the second unknown point on the given line is made to coincide with the first, so do this and then find the unknown point which minimises the total of the two straight distances.

  2. (as H. Kabayakawa) Reflect the second given point in the given line. The shortest distance between the first given point and the reflection of the first is the straight line joining them, and reflecting back the segment from the given line to the reflection of the second given point then gives the shortest path in the original question.

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    Ok. Just one comment: the fact that the straight line is the shortest curve between two points is not so trivial. But it should be known as a "fact of life".2012-07-05