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In the real Euclidean plane, let y be a circle with center A and radius of length r. Let y' be another circle with center A' and radius of length r', and let d be the distance from A to A' (see Figure 3.42). There is a hypothesis about the numbers r, r', and d that ensures that the circles y and y' intersect in two distinct points. Figure out what this hypothesis is. (Hint: Its statement that certain numbers obtained from r, r', and dare less than certain others.) What hypothesis on r, r', and d ensures that y and y' intersect in only one point, i.e., that the circles are tangent to each other?

Here is my diagram. enter image description here

What I know: m=min (r, r') M=max(r, r') d+m>M (for the y inside of y') d=M-m (for y and y' touching on the circle to left of diagram)

I've got this much but I'm not sure where to go next?

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    The biggest out of $r, r', d$ has to be equal to the sum of the remaining two. – 2017-02-26
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    `d=M-m` So you've got $d=|r-r'|$ for the case of interior tangency, which is correct. Now, derive the analog condition for exterior tangency from the figure on the left. – 2017-02-26
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    @dxiv Ok, so I have d=abs(r+r') for the analog of exterior tangency. But...do these two analog statements ensure that y and y' intersect in only one point? I feel like I'm missing something. – 2017-02-26
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    @AMerePigeon $d=r+r'$ is indeed the condition for exterior tangency (you don't need to take the absolute value since both radii are positive). The two conditions do cover all cases of tangency. To formally prove that, you could count the intersections of $AA'$ with the two circles and break it down by cases. – 2017-02-26

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