Orthocentroidal Circle
The orthocentroidal circle of a triangle 
is a central circle having the segment joining the triangle
centroid 
and orthocenter 
of 
as its
diameter (Kimberling 1998, p. 234). Since the Euler
line passes through 
and 
, it therefore
bisects the orthocentroidal circle.
It has circle function
 |
(1)
|
which corresponds to the circumcenter 
. The center of
the circle is Kimberling center 
, which has
equivalent triangle center functions
 |  |  |
(2)
|
 |  |  |
(3)
|
The circle has radius
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is the circumradius
of 
.
The circle does not pass through any notable centers other than 
and 
, which are Kimberling
centers 
and 
, respectively.
It is orthogonal to the Lester circle and Stevanović circle.
The orthocentroidal circle of any triangle always contains the incenter 
(Guinand 1984). This is an interesting
observation since it means that the incenter is always "close" to the Euler line of the triangle (although it does not lie on
it).
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area of an equilateral triangle with side length a
