Bevan point

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Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line Other points: incenter Template:Mvar, Nagel point Template:Mvar

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.Template:R

The Bevan point of a triangle is the reflection of the incenter across the circumcenter of the triangle.Template:R Bevan posed the problem of proving this in 1804, in a mathematical problem column in The Mathematical Repository.Template:R The problem was solved in 1806 by John Butterworth.Template:R

The Bevan point Template:Mvar of triangle Template:Math has the same distance from its Euler line Template:Mvar as its incenter Template:Mvar. Their distance is $$\overline{MI}=2\sqrt{R^2-\frac{abc}{a+b+c}}$$ where Template:Mvar denotes the radius of the circumcircle and Template:Mvar the sides of Template:Math.Template:R

The Bevan is point is also the midpoint of the line segment Template:Mvar connecting the Nagel point Template:Mvar and the de Longchamps point Template:Mvar.Template:R The radius of the Bevan circle is Template:Math, that is twice the radius of the circumcircle.Template:R

References

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