We have a rectangle with it sides being a mirror. A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started. Prove that at some time the light Ray passed the center of rectangle.
In a series of infinitely reflecting rectangles, a straight line can be used to represent the ray, including its reflections. In the diagram below, the *'s represent the original entry and its reflections and the O's represent the opposite corner and its reflections.
The ray of course can go directly to the opposite corner without bouncing off the walls, in which it necessarily passes through the center of the rectangle. But if it is to bounce, it must traverse an odd number of rows and an odd number (possibly 1) of columns of rectangles so the midpoint of its path is also the middle of a rectangle.
------*------+------*------+------*------+------*------
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------+------O------+------O------+------O------+------
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------*------+------*------+------*------+------*------
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------+------O------+------O------+------O------+------
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------*------+------*------+------*------+------*------
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------+------O------+------O------+------O------+------
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Posted by Charlie
on 2021-06-30 11:03:25 |
solution
