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
of 
in the circumcenter
of 
(left figure), with
 |
(1)
|
where 
is the circumradius of 
, the midpoint of the line segment joining the Nagel
point and de Longchamps point (middle figure),
as well as the reflection of the orthocenter in the
Spieker center (right figure).
The Bevan point is Kimberling center 
and has triangle
center function
 |
(2)
|
It is the center of the Bevan circle and lies on the Darboux cubic.
The Bevan point and incenter 
are equidistant from the Euler line,
both lying a distance
 |
(3)
|
away, where 
is the distance between the circumcenter and orthocenter and 
is the area of the reference
triangle (P. Moses, pers. comm., Jan. 15, 2005).
