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 of triangle has the same distance from its Euler line as its incenter . Their distance iswhere denotes the radius of the circumcircle and the sides of .
The Bevan is point is also the midpoint of the line segment connecting the Nagel point and the de Longchamps point . The radius of the Bevan circle is, that is twice the radius of the circumcircle.