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CTK Exchange
Jackie
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Dec-08-05, 07:18 PM (EST) |
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"Two Coins Problem"
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I am currently doing a project on epicycloids for Mathematics Research. I came upon the "Two Coins" problem posted on this website and I'm seeking for a proof to that problem. The solution to this problem provided is: In both motions the moving coin makes a half-turn. A half-turn around a point can be described as a reflection in that point. Choose three points A,B,C on the rolling coin. When reflected in a point O, A is carried into A' while, B and C are carried into B' and C', respectively. Another reflection in a point O', moves A', B', C' into A", B", and C", respectively. Now note, that O is the middle point of the segments AA' and BB' whereas O' is the middle point of two segments A'A" and B'B". Therefore, in both triangles AA'A" and BB'B", the line OO' connects midpoints of two sides thus being parallel and equal to half size of the third side. From here, AA" and BB" are equal and parallel (both, of course, are equal and parallel to CC" as well.) Since points A,B,C have been chosen arbitrary on the moving coin, we can now say that the whole coin was translated along a vector AA". This solution has left me puzzled. Can anyone please elaborate on this solution, especially where you claimed that "in both triangles AA'A" and BB'B", the line OO' connects midpoints of two sides thus being parallel and equal to half size of the third side." What or where is this third side? Is it possible for me to extend this "reflections" idea for two circles with unequal radii? If you have any idea, information, any solutions, or any concern or questions to this question, please email me at PHbug06@aol.com.
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alexb
Charter Member
1707 posts |
Dec-08-05, 07:33 PM (EST) |
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1. "RE: Two Coins Problem"
In response to message #0
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>This solution has left me puzzled. Can anyone please >elaborate on this solution, I'll be happy to answer your direct questions, like the one below. >... especially where you claimed >that "in both triangles AA'A" and BB'B", the line OO' >connects midpoints of two sides thus being parallel and >equal to half size of the third side." What or where is >this third side? The third side of triangle AA'A'' is AA''; that of BB'B'' is BB''. Note that we are talking of triangles AA'A'' and BB'B''. Point O is the midpoint of both AA' and BB', whereas point O' is the midpoint of sides A'A'' and B'B''. In triangle AA'A'', line OO' is the midline parallel to the side AA''. In triangle BB'B'', the same line OO' is the midline parallel to the side BB''. Since both AA'' and BB'' are parallel to OO' and have twice its length, they are parallel and equal. The above simply says that the result of two successive reflections in two points is a translation parallel to the line joining the points and to twice the distance between them. > >Is it possible for me to extend this "reflections" idea for >two circles with unequal radii? If you have any idea, >information, any solutions, or any concern or questions to >this question, please email me at PHbug06@aol.com.
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