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.

  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.

The Bevan point M of triangle ABC has the same distance from its Euler line e as its incenter I and the circumcenter O is the midpoint of the line segment MI. The length of MI is given by

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.


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