Given a triangle 
, construct the contact
triangle 
.
Now extend lines parallel to the sides of the contact
triangle from the Gergonne point. These intersect the triangle 
in the six points 
, 
,
, 
, 
,
and 
. C. Adams proved in 1843 that these
points are concyclic in a circle
now known as the Adams' circle.
Adams' circle is a central circle with circle function
 |
(1)
|
which does not correspond to any notable triangle center. Its radius is the complicated expression
 |
(2)
|
where 
is the inradius and 
is the semiperimeter of
the reference triangle and
 |
(3)
|
The center of Adams' circle is the incenter of 
(Honsberger 1995, pp. 62-74).
No notable triangle centers lie on Adams' circle.
Extend the segments 
,
, and 
to form a triangle 
. Then the Gergonne point
of 
is the symmedian
point of 
,
and Adams' circle of 
is the first Lemoine circle of 
(Honsberger 1995, p. 98).
