Bevan point

Bevan point M in triangle ABC

Bevan point M, Bevan circle k _M , height intersection H, center of gravity G, circumcenter O, incircle center I, Euler straight line e, circumference k _O

The Bevan point is one of the excellent points of a triangle . It is defined as the center of the circle that goes through the three circle centers of the given triangle. The designation Bevan point refers to a problem that the English mathematician Benjamin Bevan posed in 1806 and was solved by John Butterworth that same year.

properties

• The lines connecting the Bevan point with the circle centers are perpendicular to the sides of the given triangle.
• The connecting distance between the Bevan point and the inscribed center of the given triangle is bisected by the center of the circumference of the triangle.
• The Bevan Point is the midpoint of the line connecting the Nagel Point and Longchamps Point .
• The connecting distance between the Bevan point and the height intersection is halved by the Spieker point .
• The Bevan point and the inscribed center point have the same distance d from the Euler straight line , here the following applies ${\displaystyle d={\sqrt {R^{2}-{\tfrac {abc}{a+b+c}}}}}$
• The trilinear coordinates are . ${\displaystyle (\cos \beta +\cos \gamma -\cos \alpha -1):(\cos \gamma +\cos \alpha -\cos \beta -1):(\cos \alpha +\cos \beta -\cos \gamma -1)}$

Web links

• Eric W. Weisstein. "Bevan Point." From MathWorld - A Wolfram Web Resource
• Alexander Bogomolny. "Bevan's Point and Theorem."