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Math Help - Circles and lines! Really interesting problem!!

  1. #1
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    Red face Circles and lines! Really interesting problem!!

    Consider 3 circles with different radii. For any 2 of them one can draw 2 common tangent lines, convergent to points A, B and C.

    Prove: A, B and C are on a straight line.

    (and what makes it interesting? you ask. Well, I just want to see how many DIFFERENT proofs I can get! I have one in mind)
    Attached Thumbnails Attached Thumbnails Circles and lines! Really interesting problem!!-circles.jpg  
    Last edited by godelproof; June 20th 2011 at 08:56 AM.
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  2. #2
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    Re: Circles and lines! Really interesting problem!!

    Well... nobody responds... let me give a proof I like:

    Proof: Suppose the radii are {r}_{1},\ {r}_{2},\ {r}_{3}. Now instead of circles, imagine three BALLS with radii {r}_{1},\ {r}_{2},\ {r}_{3}. Place them in such a way that 1) centers of the balls are on a same plane (call this plane X) and 2) any ball touches the other two. Wrap 2 balls with a cone as in the attachment. For the three balls then we can wrap 3 cones. Ajust the vertex of the cones so that all of them falls on plane X. Notice that now the balls and cones are symmetric about plane X. Let plane Y be the plane that touches the 3 balls on one side of X and plane Z touches the 3 balls on the other side of X. Then the intersection \l of Y and Z falls on X, too, because of the symmetry. Clearly, A, B, C \in \l. Q.E.D.

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    The proof can be made cleaner and more rigorous. But we see the point... By escalating the problem to a higher dimension we actually make it EASIER to solve. I have this feeling not just for this particular problem: by engaging ourselves more than a problem seems to require is often a better way to work it out... paradoxically lovely idea
    Attached Thumbnails Attached Thumbnails Circles and lines! Really interesting problem!!-cone.jpg  
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  3. #3
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    Red face Re: Circles and lines! Really interesting problem!!

    Hi thank you for sharing this puzzling math problem
    I couldn't help with this though... Would love to hear other thoughts on this. Thanks!


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