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 MAMBMC of ABC
  Circumcircle of MAMBMC (Bevan circle of ABC, centered at Bevan point M)
  Reference triangle ABC
  Excentral triangle MAMBMC of ABC
  Bevan circle kM of ABC (centered at Bevan point M)
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 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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  1. ^ a b c d Encyclopedia of Triangle Centers. X(40) = BEVAN POINT
  2. ^ a b c Weisstein, Eric W. "Bevan Point". MathWorld.
  3. ^ Alexander Bogomolny. Bevan's Point and Theorem at cut-the-knot