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# Bevan Point

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).

Bevan Circle, Excentral Triangle

## 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.

Bevan Point

## Cite this as:

Weisstein, Eric W. "Bevan Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BevanPoint.html