3. The radical in the formulas indicates whether or not the circles intersect, and if they do not, whether they are both outside of each other or whether one is inside the other.

Let us assume that if the radii are not equal that

r1 > r2


r2 > 0

it follows that

2r2 > 0

so if we add   r1   to both sides,

r1 + 2r2 > r1

Subtract   r2   from both sides

r1 + r2 > r1 - r2

So there are three choices for   d.   Either

d > r1 + r2

r1 + r2 > d > r1 - r2


r1 - r2 > d.

The radicand will not be negative only in the middle case. If   d > r1 + r2   the picture will look like

If   r1 - r2 > d,   the picture will look like

There is a proof of the triangle inequality, Theorem 3.5 in this result, but it is so direct to simply use the Pythagorean Theorem, Theorem 3.4, to derive the triangle inequality that we did it that way.

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Analytic Foundations of Geometry

R. S. Wilson