geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the Brocard circle (or seven-point circle) is a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

derived from a given triangle. It passes through the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this segment is a

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

Equation

In terms of the side lengths

a

b

, and

c

of the given triangle, and the areal coordinates

(x,y,z)

for points inside the triangle (where the

x

-coordinate of a point is the area of the triangle made by that point with the side of length

a

, etc), the Brocard circle consists of the points satisfying the equation :

a^2yz+b^2zx+c^2xy=\frac\left(\frac+\frac+\frac\right).

Related points

The two

Brocard points In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...

lie on this circle, as do the vertices of the

Brocard triangle In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points construc ...

. These five points, together with the other two points on the circle (the circumcenter and symmedian), justify the name "seven-point circle". The Brocard circle is concentric with the first Lemoine circle.

Special cases

If the triangle is

equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

, the circumcenter and symmedian coincide and therefore the Brocard circle reduces to a single point.

History

The Brocard circle is named for

Henri Brocard Pierre René Jean Baptiste Henri Brocard (12 May 1845 – 16 January 1922) was a French meteorologist and mathematician, in particular a geometer. His best-known achievement is the invention and discovery of the properties of the Brocard point ...

, who presented a paper on it to the French Association for the Advancement of Science in Algiers in 1881.

References

External links