When two circles are tangent, they share a common point. For instance, consider the circles A and B.
Suppose that we wish to find a circle C that is tangent to both of these circles. In order to so, we pick two arbitrary points, one on circle A and on circle B. Then construct the radius of circle A and a line through the center of B and its arbitrary point. Build a circle by the radius of circle A with center at the arbitrary point of B, then build a triangle of the two intercepts points of the line and the built circle and the center of circle A. It is depicted as follows:
From there, find the median of the segment of AL2 , assuming the intercepts points refer to as L1 and L2.
The intersection of the line of circle B and perpendicular of segment AL2 is the locus of the two circles A and B. That locus is also the center of the circle that is tangent to both A and B. So, we obtain the following
With the locus, circle A and circle B are equidistant of the locus, regardless of were the circles are moved, which is by the way the meaning of a locus.
If observe the above figure, you might notice that the locus is located on the line of the centers of circles A and B. The reason for that is because any line tangent to a circle is perpendicular to the circle’s radius. For example, consider a line K tangent to a circle C.
If the same thing happens with a circle, the radii would have to meet at the tangent point. It is depicted as follow
Here, the line K is tangent to both circles. Therefore the circles are tangent to each other. Since, the centers of the circles are equidistant from the tangent point, the tangent point is the locus of the two circles by definition.