Let us assume that if the radii are not equal that

Since

it follows that

so if we add *r*_{1 }to both sides,

Subtract *r*_{2 }from both sides

So there are three choices for *d*. Either

*r*_{1} + *r*_{2} > *d* > *r*_{1} - *r*_{2}

or

The radicand will not be negative only in the middle case. If *d* > *r*_{1} + *r*_{2} the picture will look like

If *r*_{1} - *r*_{2} > *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.

Analytic Foundations of Geometry