Computing Homothetic Centers
For a given pair of circles, the internal and external homothetic centers may be found in various ways. In analytic geometry, the internal homothetic center is the weighted average of the centers of the circles, weighted by the opposite circle's radius – distance from center of circle to inner center is proportional to that radius, so weighting is proportional to the opposite radius. Denoting the centers of the circles and by and and their radii by and and denoting the center by this is:
The external center can be computed by the same equation, but considering one of the radii as negative; either one yields the same equation, which is:
More generally, taking both radii with the same sign (both positive or both negative) yields the inner center, while taking the radii with opposite signs (one positive and the other negative) yields the outer center. Note that the equation for the inner center is valid for any values (unless both radii zero or one is the negative of the other), but the equation for the external center requires that the radii be different, otherwise it involves division by zero.
In synthetic geometry, two parallel diameters are drawn, one for each circle; these make the same angle α with the line of centers. The lines A1A2 and B1B2 drawn through corresponding endpoints of those radii, which are homologous points, intersect each other and the line of centers at the external homothetic center. Conversely, the lines A1B2 and B1A2 drawn through one endpoint and the opposite endpoint of its counterpart intersects each other and the line of centers at the internal homothetic center.
As a limiting case of this construction, a line tangent to both circles (a bitangent line) passes through one of the homothetic centers, as it forms right angles with both the corresponding diameters, which are thus parallel; see tangent lines to two circles for details. If the circles fall on opposite sides of the line, it passes through the internal homothetic center, as in A2B1 in the figure above. Conversely, if the circles fall on the same side of the line, it passes through the external homothetic center (not pictured).
Read more about this topic: Homothetic Center, Circles
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