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
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(1)
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(2)
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(3)
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(4)
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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).
Given the above figure, 
, since
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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Because 
,
it follows that 
.
