The GEOS circle is the circle through the Kimberling centers 
(de Longchamps point 
; intersection 
of the Soddy
line and Euler line), 
(intersection 
of the Euler
line and orthic axis), 
(intersection 
of the Gergonne
line and orthic axis), and 
(Fletcher point;
intersection 
of the Gergonne line and Soddy
line) (F. Jackson, pers. comm., Oct. 20, 2005).
The GEOS circle has a rather complicated radius. Its center is the midpoint of 
and 
, which has triangle
center function
 |
where 
and 
, 
, 
,
and 
is Conway
triangle notation (P. Moses, pers. comm., Oct. 20, 2005), which is
not a Kimberling center.
It has the simple circle function
 |
which also does not correspond to any Kimberling center.
The four lines also determine a companion circle through 
, 
, 
, and 
. Its diameter
is 
. Moreover, 
, 
, 
, and 
form an orthocentric
system. These statements assume that the indicated intersections are finite and
distinct (P. Moses, pers. comm., Nov. 2, 2005; F. M. Jackson,
pers. comm., Dec. 5, 2005).
