Let $C_1$ and $C_2$ be the centres of two circles, $r_1$ and $r_2$ be their radii respectively. I know that the condition for the existence of four common tangents between the two circles is $r_1 + r_2 < C_1C_2$. Similarly, the condition for only three common tangents is $r_1 + r_2 = C_1C_2$. Both these conditions are pretty obvious.
Now, the condition for the existence of (only) two common tangents between the two circles is $|r_1 - r_2| < C_1C_2<r_1 + r_2$, and the condition for no common tangents is $C_1C_2 < |r_1 - r_2|$. How are these two conditions derived?
