Consider 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). Amazingly, these points are concyclic in a circle here dubbed the GEOS
circle (F. Jackson, pers. comm., Oct. 20, 2005).
The GEOS circle has rather complicated radius. Its center is the midpoint of 
and 
, which has 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.
By definition, the GEOS circle passes through Kimberling centers 
for 
(de Longchamps point),
468, 650, and 1323 (Fletcher point).
