Bevan point

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.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{} Reference triangle ^ABC Excentral triangle ^MAMBMC of ^ABC Circumcircle of ^MAMBMC (Bevan circle of ^ABC, centered at Bevan point M) Bevan punkt.svg
  Reference triangle △ABC
   Excentral triangle △MAMBMC of △ABC
  Circumcircle of △MAMBMC (Bevan circle of △ABC, centered at Bevan pointM)
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 Bevan2 punkt.svg
  Reference triangle △ABC
   Excentral triangle △MAMBMC of △ABC
  Bevan circlekM of △ABC (centered at Bevan pointM)
   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. [1]

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

  1. 1 2 3 4 Encyclopedia of Triangle Centers. X(40) = BEVAN POINT
  2. 1 2 3 Weisstein, Eric W. "Bevan Point". MathWorld .
  3. ↑ Alexander Bogomolny. Bevan's Point and Theorem at cut-the-knot


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