The Bevan point 
of a triangle 
is the circumcenter of the excentral
triangle 
. It is named in honor of Benjamin Bevan,
a relatively unknown Englishman proposed the problem of proving that the circumcenter 
was the midpoint
of the incenter 
and the circumcenter of the
excentral triangle and that the circumradius of the excentral triangle was 
(Bevan 1806), a problem solved by John Butterworth (1806).
|
|
|
is the reflection
of the incenter 
in the circumcenter 
(left figure), with
 |
(1)
|
where 
is the circumradius and 
, 
, and 
are the side lengths
of 
. It is also the midpoint
of the line segment joining the Nagel
point 
and de Longchamps point 
(middle figure), and the reflection
of the orthocenter 
in the Spieker center 
(right figure). The corresponding squared
distances are
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the inradius and 
is the semiperimeter (Queiroz
Filho 2022).
The Bevan point is Kimberling center 
and has triangle
center function
 |
(4)
|
It is the center of the Bevan circle and lies on the Darboux cubic.
For 
, the Bevan point and incenter 
are equidistant from the Euler
line (P. Moses, pers. comm., Jan. 15, 2005). Their common distance
is
 |
(5)
|
where 
is the distance between the circumcenter
and orthocenter, and 
is the area of the reference
triangle.
