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Extangents Circle


ExtangentsCircle

The extangents circle is the circumcircle of the extangents triangle. Its center function is a complicated 9th-order polynomial and its circle function is a complicated 6th-order polynomial. Its center lies on the lines (5, 19), (26, 55), and (30, 40), and is therefore is on a line parallel to the Euler line through X_(40).

Its radius however is given by the nice expression

 R_E=(a^2b^2c^2[S^2+abc(a+b+c)])/(4(a+b+c)SS_AS_BS_C),

where S, S_A, S_B, and S_C are Conway triangle notation (P. Moses, pers. comm., Jan. 15, 2005).

No Kimberling centers lie on the extangents 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, Extangent, Extangents Triangle, Intangents Circle

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Cite this as:

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

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