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 
.
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),](/images/equations/ExtangentsCircle/NumberedEquation1.svg) |
where 
,
, 
, and 
are Conway triangle
notation (P. Moses, pers. comm., Jan. 15, 2005).
No Kimberling centers lie on the extangents circle.
Let 
be the center of the intangents circle and 
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 (
and 
,
respectively). Amazingly, the midpoint of 
is the circumcenter of the tangential triangle 
, which lies on the Euler line
(P. Moses, pers. comm., Jan. 15, 2005).
