geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

derived from a given triangle. It passes through the

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

and

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

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 centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

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 :

b^2 c^2 x^2 + a^2 c^2 y^2 + a^2 b^2 z^2 - a^4 y z - b^4 x z - c^4 x y=0.

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 with sides , where the vertices are labeled in counterclockwise order, ther ...

lie on this circle, as do the vertices of the Brocard triangle. 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 An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

, 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 p ...

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

Algiers Algiers is the capital city of Algeria as well as the capital of the Algiers Province; it extends over many Communes of Algeria, communes without having its own separate governing body. With 2,988,145 residents in 2008Census 14 April 2008: Offi ...

in 1881.

References

External links