Set up a coordinate system with origin at the center of one circle, with radius r, and the positive x-axis through the center of the other circle, with radius R. Let be the coordinates of the center of the second circle. Taking as the angle the two radii to the points of tangency make with the positive x-axis (because they are parallel this angle is the same for both radii), We can write the coordinates of the point of tangency of the first circle as and and the coordinates of the point of tangency of the second circle as and .
Then and . Therefore the slope of the mutual tangent line is . Since that is perpendicular to the radius from the center of the first circle to the point of tangency, which has slope , we must have
"Cross multiplying", we have and, using the fact that , that reduces to or, finally, .
(If , then the the smaller circle would be inside the larger so there would be no mutual tangent.)
We now have and . It is the fact that we can use the positive or negative root that gives two different y values for the same x value and the two different points of "mutual tangency" on the first circle.
Use and to get the two points of tangency on the other circle.