Bevan Point


The Bevan point V of a triangle DeltaABC is the circumcenter of the excentral triangle DeltaJ_AJ_BJ_C. It is named in honor of Benjamin Bevan, a relatively unknown Englishman proposed the problem of proving that the circumcenter O was the midpoint of the incenter I and the circumcenter of the excentral triangle and that the circumradius of the excentral triangle was 2R (Bevan 1806), a problem solved by John Butterworth (1806).


V is the reflection of the incenter of DeltaABC in the circumcenter of DeltaABC (left figure), with


where R is the circumradius of DeltaABC, 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 X_(40) and has triangle center function


It is the center of the Bevan circle and lies on the Darboux cubic.


The Bevan point and incenter I are equidistant from the Euler line, both lying a distance


away, where |OH| is the distance between the circumcenter and orthocenter and Delta is the area of the reference triangle (P. Moses, pers. comm., Jan. 15, 2005).

See also

Bevan Circle, Excentral Triangle

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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.

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