Bevan's theorem for a triangle states that if 
and 
are the incenter and circumcenter
of a reference triangle 
, and 
is the circumcenter of its
excentral triangle, then 
, 
,
and 
are collinear with 
, and the circumradius
of the excentral triangle is 
, where 
is the circumradius of 
.
Indeed, 
is the orthocenter of the excentral
triangle, while 
is its orthic triangle. The circumcircle
of 
is therefore the nine-point circle of the excentral triangle, with 
as its nine-point center.
Since a nine-point center is the midpoint
of the orthocenter and circumcenter,
is the midpoint
of 
and 
.
The nine-point circle has half the circumradius
of the excentral triangle, giving the second
conclusion.
