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Circle-Circle Tangents

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).

Given the above figure, , since

 (5) (6) (7) (8) (9) (10) (11) (12)

Because , it follows that .

Circle Tangent Line, Descartes Circle Theorem, Extangent, External Similitude Center, Eyeball Theorem, Homothetic Center, Intangent, Internal Similitude Center, Midcircle, Miquel Point, Monge's Circle Theorem, Monge's Problem, Nine-Point Circle, Pedal Circle, Tangent Circles, Tangent Line, Triangle

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., 1888.Dixon, R. Mathographics. New York: Dover, p. 21, 1991.Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 4-5, 1991.

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Circle-Circle Tangents

Cite this as:

Weisstein, Eric W. "Circle-Circle Tangents." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circle-CircleTangents.html