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Intangents Circle


IntangentsCircle

The intangents circle is the circumcircle of the intangents triangle.

It has circle function

 l=((-a+b+c)f(a,b,c))/(8a^2b^2c^2cosAcosBcosC),
(1)

where

 f(a,b,c)=a^5-a^3b^2-a^2b^3+b^5+a^3bc+a^2b^2c-ab^3c-b^4c-a^3c^2+a^2bc^2+2ab^2c^2-a^2c^3-abc^3-bc^4+c^5,
(2)

which is not a Kimberling center.

Its center has center function

 alpha=-a^6+a^4b^2+a^2b^4-b^6+2a^4bc-a^2b^3c-b^5c+a^4c^2-2a^2b^2c^2+b^4c^2-a^2bc^3+2b^3c^3+a^2c^4+b^2c^4-bc^5-c^6,
(3)

and it has radius

 R_I=r/(4|cosAcosBcosC|),
(4)

where r is the inradius of the reference triangle.

No Kimberling centers lie on the intangents circle.

Let O_I be the center of the intangents circle and O_J the center of the extangents circle. Both centers lie on the line (26, 55), and both centers are on lines parallel to the Euler line through simple points (X_1 and X_(40), respectively). Amazingly, the midpoint of O_IO_J is the circumcenter of the tangential triangle X_(26), which lies on the Euler line (P. Moses, pers. comm., Jan. 15, 2005).


See also

Central Circle, Extangents Circle, Intangent, Intangents Triangle

Explore with Wolfram|Alpha

Cite this as:

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

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