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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

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

where is the inradius and is the semiperimeter (Queiroz Filho 2022).

The Bevan point is Kimberling center and has triangle center function

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

where is the distance between the circumcenter and orthocenter, and is the area of the reference triangle.

See also

Bevan Circle, Bevan's Theorem, Excentral Triangle

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References

Bevan, B. "VII. Question 67." In New Series of the Mathematical Repository, Vol. 1 (Ed. T. Leybourn). London, England: W. Glendenning, p. 18, 1806.Bogomolny, A. "Bevan's Point and Theorem." https://cut-the-knot.org/Curriculum/Geometry/BevanPoint.shtml.Butterworth, J. New Series of the Mathematical Repository, Vol. 1 (Ed. T. Leybourn). London, England: W. Glendenning, p. 143, 1806.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Queiroz Filho, J. F. de. "The Bevan's Point and Its Special Relationships." Romanian Math. Mag. No. 1, 4-17, 2022.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 159, 1991.

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Bevan Point

Cite this as:

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

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