Bevan point

  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

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.

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

$${\overline {MI}}=2{\sqrt {R^{2}-{\frac {abc}{a+b+c}}}}$$

where R denotes the radius of the circumcircle and a, b, c the sides of △ABC. 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.

External links

• Eric W. Weisstein. Bevan Point. From MathWorld--A Wolfram Web Resource
• Alexander Bogomolny. Bevan's Point and Theorem at cut-the-knot
• Encyclopedia of Triangle Centers. X(40) = BEVAN POINT