In geometry, the Bevan point, named after Benjamin Bevan, is a triangle center. It is defined as center of the Bevan circle, that is the circle through the centers of the three excircles of a triangle. The Bevan point of a triangle is the reflection of the incenter across the circumcenter of the triangle. Bevan posed the problem of proving this in 1804, in a mathematical problem column in The Mathematical Repository. The problem was solved in 1806 by John Butterworth. The Bevan point M of triangle △ABC has the same distance from its Euler line e as its incenter I. Their distance is
M I ¯ = 2 R 2 − a b c a + b + c {\displaystyle {\overline {MI}}=2{\sqrt {R^{2}-{\frac {abc}{a+b+c}}}}}
where R denotes the radius of the circumcircle and a, b, c the sides of △ABC. The Bevan is point is also the midpoint of the line segment NL connecting the Nagel point N and the de Longchamps point L. The radius of the Bevan circle is 2R, that is twice the radius of the circumcircle.
References