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).

See also Bevan Circle ,

Excentral
Triangle
Explore with Wolfram|Alpha
References Bevan, B. "VII. Question 67." In New Series of the Mathematical Repository, Vol. 1 (Ed. T. Leybourn).
London: W. Glendenning, p. 18, 1806. Butterworth, J. New
Series of the Mathematical Repository, Vol. 1 (Ed. T. Leybourn).
London: W. Glendenning, p. 143, 1806. Kimberling, C. "Triangle
Centers and Central Triangles." Congr. Numer. 129 , 1-295, 1998. Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
p. 159, 1991. Referenced on Wolfram|Alpha Bevan Point
Cite this as:
Weisstein, Eric W. "Bevan Point." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/BevanPoint.html

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