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Bevan point

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Triangle center: circumcenter of a triangle's excentral 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)   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. Bevan posed the problem of proving this in 1804, in a mathematical problem column in The Mathematical Repository. The problem was solved in 1806 by John Butterworth.

The Bevan point M of triangle △ABC has the same distance from its Euler line e as its incenter I. Their distance is M I ¯ = 2 R 2 − a b c a + b + c {\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.

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.

References

  1. ^ Encyclopedia of Triangle Centers. X(40) = BEVAN POINT
  2. ^ Weisstein, Eric W. "Bevan Point". MathWorld.
  3. Alexander Bogomolny. Bevan's Point and Theorem at cut-the-knot


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