Given two circles with the radius of $1$, two circles with a radius of $2$ and two coins of the radius $3$. It is allowed to put two of them so these circles would touch each other.. Then the circles are constructed one by one,the added circle must touch at least two of the other circles on the plane. Circles may not overlap. Is it possible to position the circles so that the centers of three any coins would locate on one line?
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$\begingroup$ If the circles may overlap, yes, for example r1 and r2 are inside r3. Then place another r1 inside r2 so it touches the first r1. Then place an r2 at one and an r3 at the other end. Each touches exactly 2 others. $\endgroup$– Weather VaneCommented Oct 22, 2017 at 10:28
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$\begingroup$ Forgot to mention it in the problem, sorry. It is not allowed to overlap circles. $\endgroup$– student28Commented Oct 22, 2017 at 10:30
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1$\begingroup$ That makes the question rather silly. If the first two circles must touch, it is impossible to place another on the same line so it touches them both. $\endgroup$– Weather VaneCommented Oct 22, 2017 at 10:32
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$\begingroup$ Is it allowed for one circle to be inside another? $\endgroup$– Dan UznanskiCommented Oct 22, 2017 at 10:34
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$\begingroup$ That was my answer. $\endgroup$– Weather VaneCommented Oct 22, 2017 at 10:34
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Is this solution valid? I don't think there is a solution unless some circles are included in others
answered Oct 22, 2017 at 10:49