669. The angle at which two circles intersect is called the angle formed by the tangents to these circles at the point of intersection. When do two circles intersect at an angle of $0^{\circ}$? $90^{\circ}$? $180^{\circ}$?
669. $0^{\circ}$ at internal tangency of circles, $180^{\circ}$ at external tangency of circles (gears rotate in the same direction when in internal engagement, and in opposite directions when in external engagement). When intersecting at a right angle, the tangents at the point of intersection, and therefore the radii drawn to this point, form a right angle. Therefore, if $c$ denotes the distance between the centers, $\boldsymbol{R}$ and $\boldsymbol{r}$ the radii of the circles:

$$
\begin{aligned}
& \begin{array}{lll}
\text { for } c=\boldsymbol{R}-\boldsymbol{r} & \text { the angle of intersection of the circles } \quad 0^{\circ}
\end{array} \\
& \text { for } c=\boldsymbol{R}+\boldsymbol{r} \quad \text { » } \quad » 180^{\circ} \\
& » \quad \boldsymbol{c}=\sqrt{\boldsymbol{R}^{2}+\boldsymbol{r}^{2}} \quad \geqslant \quad \geqslant 90^{\circ}
\end{aligned}
$$