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Math Help - Proof on Interior Points + Pool Table

  1. #1
    Senior Member MacstersUndead's Avatar
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    Proof on Interior Points + Pool Table

    The question is
    "Prove that an interior point of a triangular pool table cannot be invisible."

    pool table: a ray will bounce off a side such that angle of incidence = angle of reflection. If a ray hits a corner, the path ends there.

    invisible: A point A in a polygon is called an invisible point if there is no pool shot from A coming back to A.
    --

    I'm thinking it is a proof by contradiction with some use of the Unfolding Principle. (Given a pool shot in some polygon P then we can unfold the pool shot to a straight line path by taking appropriate mirror images in the sides of P.
    EDIT:// is it true that if a path doesn't include P, it would have to periodically not include P?

    any sort of help will be appreciated. thank you in advance.
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by MacstersUndead View Post
    The question is
    "Prove that an interior point of a triangular pool table cannot be invisible."

    pool table: a ray will bounce off a side such that angle of incidence = angle of reflection. If a ray hits a corner, the path ends there.

    invisible: A point A in a polygon is called an invisible point if there is no pool shot from A coming back to A.
    --

    I'm thinking it is a proof by contradiction with some use of the Unfolding Principle. (Given a pool shot in some polygon P then we can unfold the pool shot to a straight line path by taking appropriate mirror images in the sides of P.
    EDIT:// is it true that if a path doesn't include P, it would have to periodically not include P?

    any sort of help will be appreciated. thank you in advance.
    Maybe I'm missing something, but isn't it always possible to go from point A to one of the sides of the triangle such that the angle of incidence is 0 degrees (ie, the ray and the side are perpendicular), thus getting you back to A in one bounce?
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  3. #3
    Senior Member MacstersUndead's Avatar
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    Interesting! I never thought about it like that. I don't think there's a counterexample for it... thank you!

    EDIT: I tried to construct one with a scalene triangle, but still it works out.
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  4. #4
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    This is really wonderful issue, i have never seen before. Thanks for giving me such a useful and informative posting. I know a company who manufacture pool and billiards table. Hope this helps in future.
    www.chevillotte.com
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