Bevan point Reference triangle △ABC Circumcircle of △MAMBMC (Bevan circle of △ABC, centered at Bevan point M)  Reference triangle △ABC Bevan circle kM of △ABC (centered at Bevan point M) Other points: incenter I, Nagel point N 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.[1] The Bevan point of a triangle is the reflection of the incenter across the circumcenter of the triangle.[1] Bevan posed the problem of proving this in 1804, in a mathematical problem column in The Mathematical Repository.[1][2] The problem was solved in 1806 by John Butterworth.[2] The Bevan point M of triangle △ABC has the same distance from its Euler line e as its incenter I. Their distance is where R denotes the radius of the circumcircle and a, b, c the sides of △ABC.[2] The Bevan is point is also the midpoint of the line segment NL connecting the Nagel point N and the de Longchamps point L.[1] The radius of the Bevan circle is 2R, that is twice the radius of the circumcircle.[3] References
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