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I have a test tomorrow and this might be a question on it. I do not know the answer and I have no idea how I would draw it out if I had to. The question is...

Describe the locus of the centers of all circles of the same size that are tangent to a given line.

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    $\begingroup$ Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. $\endgroup$ – Pedro Tamaroff Apr 12 '13 at 0:57
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Consider a person riding a bike. The circle is the wheel, the line is the floor. The center is the bolt with which the wheel attaches to the bike. How will the bolt move if the floor keeps perfectly flat and even? Will it's height relative to the floor change? As the bike moves then, what will the bolt describe? Making a drawing should prove very useful.

ADD It might also be the case you're being asked to consider both sides of the line. Then you'll just get a mirrored version of what you're getting from the above.

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  • $\begingroup$ Would the drawing be a line with a bunch of circles above it? $\endgroup$ – Chris Apr 12 '13 at 1:03
  • $\begingroup$ @Chris Yes, but really, you should care about what the bolt describes, not the circles. Make a few circles and connect the dots. Alternatively, suppose we put a light on the bolt and took a prolonged exposure shot after the bike moved for a while. $\endgroup$ – Pedro Tamaroff Apr 12 '13 at 1:05
  • $\begingroup$ Is the locus then a line parallel to the tangent directly above the circle. $\endgroup$ – Chris Apr 12 '13 at 1:10
  • $\begingroup$ @Chris You got it almost right. The line must be "inside" the circle, since the center is. Can you see that? $\endgroup$ – Pedro Tamaroff Apr 12 '13 at 1:15
  • $\begingroup$ So the line is at the midpoints of all the circles $\endgroup$ – Chris Apr 12 '13 at 1:18

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