An elementary (?) minimization problem 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.
 A: 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: 


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*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.

*(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. 
