Bevan point

[[File:Bevan punkt.svg|thumb|upright=1.0|

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Other points: incenter $M$, Nagel point $kM$ ]]

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 $O$. Their distance is $$\overline{MI} = 2\sqrt{R^2-\frac{abc}{a+b+c}} $$ where $H$ denotes the radius of the circumcircle and $G$ the sides of $△MAMBMC$.

The Bevan is point is also the midpoint of the line segment $L$ connecting the Nagel point $I$ and the de Longchamps point $N$. The radius of the Bevan circle is $△ABC$, that is twice the radius of the circumcircle.