The points of tangency and for the four lines tangent to two circles with centers and and radii and are given by solving the simultaneous equations

(1)

(2)

(3)

(4)

The point of intersection of the two crossing tangents is called the internal similitude center. The point of intersection of the extensions of the other two
tangents is called the external similitude
center.

Therefore, for a given triangle , there are four lines simultaneously tangent to the
incircle and the -excircle. Of these, three correspond
to the sidelines of the triangle, and the fourth is known as the -intangent. Similarly, there
are four lines simultaneously tangent to the - and -excircles. Of these, three correspond
to the sidelines of the triangle, and the fourth is known as the -extangent.

A line tangent to two given circles at centers and of radii and may be constructed by constructing the tangent to
the single circle of radius centered at and through , then translating this line along the radius through
a distance
until it falls on the original two circles (Casey 1888, pp. 31-32).