Bevan point

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]

  Reference triangle △ABC

  Excentral triangle △M_AM_BM_C of △ABC

  Circumcircle of △M_AM_BM_C (Bevan circle of △ABC, centered at Bevan point M)

  Reference triangle △ABC

  Excentral triangle △M_AM_BM_C of △ABC

  Bevan circle k_M of △ABC (centered at Bevan point M)

  Euler line e, on which circumcenter O, orthocenter H, centroid G, and de Longchamps point L lie

Other points: incenter I, Nagel point N

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 $${\displaystyle {\overline {MI}}=2{\sqrt {R^{2}-{\frac {abc}{a+b+c}}}}}$$ where R denotes the radius of the circumcircle and a, b, c the sides of △ABC.^[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]