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
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).